Flexibility is great.

I play BLM (ilvl 85) and MNK (ilvl 90) almost equally at this point (Turn 1 and 5 as MNK, Turn 2 and 4 as BLM).
If I do Garuda, I go as a BLM and if I do Titan EM or Ifrit EM I go as a MNK.
I also do lower content than that (HMs, CT, Mog) as a WHM. Faster queues are nice for CT.
So that's my personal experience and why I would vouch for multiple characters.

Though the armoury system gets a lot of flak. I really appreciate that I can get gear for other classes on one character.
That's not the case in other games. When I gave SMN a try, I had quite a few allagan pieces that transferred over to it.
When I got my WHM to 50, I had some 90 waiting as well without spending any myth.

So even with the limitations, there are still quite a few benefits. Especially since non-myth gear can fit multiple classes (except bard/archer), so as you keep playing it's increasingly easier to try out multiple classes with better gear.

So gearing up doesn't seem that slow to me.
And with that in mind, levelling multiple classes in this game would be for the same reason as any other game -
- Flexibility
- Trying out different playstyle
- Understanding how the other classes you play with work from a first person perspective