Been a while since anyone posted here but, since this is in the Crafting area of the Forum and I've been thinking about just this question, I'll have a go at it.
For the TLDR crowd out there: FFXIV crafting is fun, it's one of the best in current MMOs and, yes, it makes sense to do all the classes on one character if you are REALLY into crafting.
Crafting in FFXIV is one of the richest crafting games in any of the MMOs I've played, and I've played about a dozen in the last twelve years -- from SWG and EVE to SWTOR, and Vanguard and LOTRO for more of a fantasy setting. I think that the most central aspect of any gaming experience is always going to be how results are generated from random chance and, most importantly, what options a game gives you for modifying that chance or "mitigating the chance for failure" to get the results you want. Also central to a meaningful crafting experience is being able to craft things players will need and use ... not just one or two things, but the majority of what you can make.
Science fiction and fantasy games should approach crafting differently. Scifi is about the technology, and so the advantages that tech would offer should play a role. Star Wars Galaxies was a great example of how that could be played out. SWG had an incredibly rich, complex method for generating the quality of the items you could craft. While you could craft items one at a time, the most efficient and profitable focus to take was to craft schematics that could be loaded into factories to generate multiple copies of an item. Whether crafting an item or a schematic, though, the process was the same. There was always a chance for failure as well as a chance to generate items of high quality. You modified your chances in crafting through skills, through gear, through buffs from food and entertainers and city specializations, until you could just about guarantee high-quality successes for what you did.
When SWG was still young, there were many features about the game that made crafting central to everyone's experiences. 99% of the gear you needed was made by player crafters, with only some of the rarest items being things that were looted. Even then, things like jetpacks or Mandalorian armor still required some crafting to be done. As the game changed, crafting's importance became much more watered down ... items no longer decayed, crafting dynamics were simplified. At one point (at the start of the NGE), crafting almost became pointless and unnecessary. It did make a bit of comeback towards the end.
The thing most SWG players think most about for crafting was the "resource game". There were dozens and dozens of resources, subtypes to every single kind of material you needed (seven types of steels and most inorganics, organics based on planets, etc.). Resources quality varied: a particular type of steel at one given moment would be identified with a unique name and unique set of stats, and within one to two weeks of its appearance it would "shift" out of availablity. It might be replaced immediately by a new variation of the same steel with its own unique name and unique set of stats; it might take a few days or weeks for it to reappear. While you could always "hand sample" to get small quantities of anything, once you found something you needed you would place automated harvesters where you could find that resource to gather mass quantities of it, which then allowed you to mass produce items made from it with your factories. The resource game in SWG is what created the greatest amount of variation in gear quality and investing in resource management was critical to being a successful crafter: it was possibly both the greatest attraction to crafting as well as the greatest barrier to becoming a successful crafter.
Sorry for the long-winded start -- SWG players loved (like me) or hating crafting. Either way, it still is probably the most complex crafting system any MMO has seen, and that complexity both makes it great and makes it a perfect example of what NOT to do for games that followed it. For many, many players, it simply was too complex, too "grindy". It's also a great example of what fantasy MMO crafting systems should NOT be, even though it was a great system. Fantasy is dealing with low tech, but it also deals with things like magic or alchemy that have no role in (hard) scifi games.
As for fantasy games, crafting in FFXIV reminds me a lot of the system in Vanguard -- which is still running as a F2P game through Sony Online Entertainment. Resource quality is simplified into two or three levels of quality for each type of resource. Resources vary in "level" similar to the combat levels in the game, to which the crafting levels are matched as well. Each item you learn to craft starts out being difficult to even complete but, as you gain experience and get better gear, you become more capable of crafting any item at a specific difficulty level. The crafting process also involves two lines of progress: progress towards item completion and improvements in item quality with two or three possible outcomes for item quality. Individual crafting skills you learn allow you to alter progress or quality or both. A major difference between Vanguard and FFXIV is that Vanguard allows for complex crafting errors -- separate from failing at any step is the chance for a complication to arise. Complications can reduce the effectiveness of your abilities and even reverse the progress you have made unless you take quick actions to remedy those complications. It's an interesting dynamic to the crafting process that FFXIV does not have but, overall, it can seem as much of a detraction to the process as an additional "puzzle" to be solved in item creation.
As for FFXIV specifically? As I said at the start, what matters in terms of game mechanics is how your chance to succeed is randomized and, more importantly, how your skills allow you to modify your chances -- mitigating any chance of failure as well as improving your chances of creating exceptional items. At first glance, FFXIV crafting might seem like a simple system: you have two types of crafting resources (normal and high quality), you have two progress bars that need your attention while crafting (progress and chance for HQ), and you have two possible successful results (normal and HQ). If you don't care about quality, it's extremely difficult to FAIL at creating an item.
If you've tried your hand at crafting in FFXIV, however, you know things aren't that simple. There are two major reasons for this, from my perspective. Perhaps most importantly, what you can craft is precisely what people need to play the game. Sure, quests give gear as rewards. The majority of it, though, is identical to normal quality gear that can be crafted. High quality crafted gear is generally superior to the "green" loot drops from treasure chests. Even the pink "Aetherial" gear drops are only sometimes better than HQ crafted gear, and that would be in it having one additional stat. For example, the Aetherial loot drop version of a HQ crafted item will have the same set of stats and, most likely but not certainly, one additional stat such as Skill Speed or Spell Speed. Crafted gear, on the other hand, have slots for materia melding that loot drops do not have, which means that the HQ crafted versions are more adaptable or customizable to the specific stats you WANT to have. One last point here -- I certainly have not played all there is to play yet, but I haven't seen any HQ food or potions as loot drops or quest rewards. Correct me if I'm wrong, but it appears that the only way to get HQ versions of these are from crafters.
The fact that players WANT crafted gear and items means that your decision to craft HQ items is a meaningful, important decision. Going for 100% chance of HQ isn't just a puzzle to be solved; it's a way to earn a living. That makes the skills each crafting class has available to it meaningful as well. And this is where FFXIV's crafting system really does shine. As you progress in level, you gain greater options in your Synthesis and Touch skills that make you choose between certainty of success or amount of progress towards completion/quality improvements. In addition to these primary actions, you have a host of others that can alter just about every aspect of the crafting process: reducing or reversing reductions in durability, multiplying amounts of successful progress, modifying chance for success or item difficulty.
The most important, compelling thing about crafting in FFXIV is this: you need more "skill" than the actions your class(es) give you -- you need to understand what they do and know when to use each one. That's not any "skill" that pushing a button in the game gives you; it's a skill that's inside your head.
And to toss in one more twist, no one particular class has all the actions you need to be successful. Maybe someone out there has played a game with a class system similar to FFXIV, but I haven't. As someone who loves the crafting game, I'm one of those people who has to be able to do everything. When I was playing SWG, that meant having something like three or four separate accounts in order to be able to craft everything. In SWTOR, that means logging out of one toon and into any one of five others in order to craft what I want, since there are six crafting professions and each toon can have only one. When I first started playing FFXIV and realized you could have not just more than one class for any one toon, you could level up every single class on the same toon, I thought why? I want to play different races, different genders, different looks, different city storylines. Then I noticed that "Additional" tab on the Actions & Traits window. Then I read something about how those different skills for different classes were NOT just the same abilities with different names. All of a sudden, the ability to play different crafting (and harvesting and combat) classes became a very real need for me. Do you need to be a "Builder of the Realm" to be a successful crafter? No. But if you look into what skills the different classes have with Affinities for Disciples of the Hand, you will see that each class offers skills that ALL classes can use, and you get them as early as level 15, an easily accomplished goal. That you can "do it all" on one character is interesting, as well as something I have not seen in any other game. That doing it all on one character makes each class that much more powerful is something that really makes this aspect of the game shine.
One last MMO I want to talk about: SWTOR. Crafting in SWTOR is everything an MMO needs to have the WORST in the business. It's absolutely dreadful. Why? Primarily, there is nothing that can be crafted that is not as good as what you can get from running repeatable instances. For pvp, there is nothing craftable that is necessary. For pve, you can "reverse-engineer" certain loot drops from endgame instances to learn how to craft certain items, but the only reason anyone would want to actually buy these is for convenience. To have it now, instead of waiting to win a loot roll. Because nothing crafted is needed, nothing crafted is valued. SWTOR is rife with people giving away the only desirable crafted items in trade for mats. Crafters don't even ask for tips. And because there is a small, random chance in crafting these items that the results will be doubled (two of an item instead of one), players see no need to pay crafters beyond trading needed materials (which come as loot drops in endgame instances). Crafting is completely unnecessary.
Now, to twist the knife a bit more: even if you want to craft, you do not actually craft anything. Crafting is a "crew skill". If you're not familiar with SWTOR, you gain a crew of NPC companions who travel with you and fight with you, kind of like your chocobo. You can only have one out with you at once, but you can set the others to doing "skills", Each companion actually has very specific things it's good at as well. Here's the rub: when it comes to crafting, your companions have, at best, an 80% chance to FAIL any critical success you need. Given companion abilities and affection for you (you can bribe them with gifts ... do chocobos like apples?), it is possible in rare cases to improve this to a 70% chance to fail in getting the result you want. At worst, you will have a 90% chance to fail. And there is nothing you can do about it. Crew "skills" have nothing to do with real skill. No matter how much you know about the game, BioWare simply has not provided any means of exercising that knowledge into mitigating that atrociously huge chance to fail. Your life as a crafter in SWTOR is writ large in your failures, and there is nothing you can do to change that. The smartest choice you can make in SWTOR about crafting is to not craft at all.