There's three big reasons SE is reluctant to do this, some of which have been covered thoroughly in this thread:
- LORE: A huge part of this game is the conflict between the Human races and the Beast races. The player storyline has them pretty thoroughly entrenched as a Human member of this conflict. If a bestial race was added, they'd either need to retool the storyline, or explain why this new race is somehow considered to be a Human race, in spite of the fact that they look no more human than the Amal'ja.
- EQUIPMENT: Any new race needs to be able to wear all of the equipment in the game. Human-looking races require only minor tweaks to the thousands of items in the game - but depending on how bestial the new race is (for example, digitigrade feet, muzzles) all these pieces of gear would need extensive redesigns.
- POPULARITY: We've already seen what SE's perception of player reaction did to those poor Au Ra girls. It's unlikely they'll take such an extreme step as an outright Beast race.

From a personal standpoint, I'm fine with the current status quo. FFXIV's roster of all-human races is part of its identity as a game. I'd also like to avoid the tired old trope that pops up in a lot of RPGs that features standard fantasy races (Human, Elf, Dwarf, etc) and one bizarre, off-the-wall race (Squiddoid, or whatever). If a game is going to mix humans and inhumans, it should be built that way from the ground up, not try to shoehorn them in later.