There are tons of reasons not to use alts in this game.
  1. There are no crafted endgame gear items. Crafted items end about 20 ilevels short of the endgame items. Also, to do endgame crafts, you need nearly every crafter class at 50 for their level 50 cross-class skills.
  2. Many endgame equipment items can be shared across similar classes, like BLM and SMN.
  3. You get a huge experience bonus if you're leveling a new class and already have one at 50 (like, +50%).
  4. Once you hit 50 and stop gaining experience, you still get "Rested" experience from being in town or logging out in town. This means that when you do switch to a new class to level it, you get another +50% bonus to your experience (so, double exp, when combined with the above).
  5. Cross-class skills are a thing. For each job, there are two other classes that have skills you can use as that job. The last of these skills tend to be learned around level 26-42, so you don't have to get the other classes to 50 to learn all the cross-class skills, but you'll have to do tons of leveling of the same classes if you want to have an alt for each job or whatever.
  6. Flexibility. Out grinding Atma? WHM is a terrible class for soloing low-level FATEs, so you probably want to switch to a DPS class. Doing some Spirit-bonding? You want to do massive AoE damage fast, so switch to WHM (they get Holy at endgame) or BLM. Did an S-rank Hunt mob just spawn? You'll want to Tank or Heal that, but switching from your DPS character to a Tank or Heal character takes time, and the mark will be dead by the time you switch characters and get there.
  7. More importantly, want to queue for a dungeon as a DPS, with a queue time of ~30 minutes, and not be bored to death while you wait? Sure, you could FATE grind or whatever, but it's really nice to have a class at endgame so you have something to do while you queue, like Beast Tribe dailies or Spiritbonding, or a crafter/gatherer class to level up.

Really, the biggest reason to do it all on one character is convenience. You don't have to take up 8 spots in an FC. You don't have to manage your macros and equipment between characters. You can use dungeon drops you get for other classes (99% of which are untradeable). You can get hard-to-obtain vanity items for glamours and use them for any class instead of just one. You don't have to level 8 different chocobos. You can easily switch classes if you want to do a given content, but your class isn't needed.

And so on.