The pets are just a special reward for doing the dungeon. Keeping them rare is a tool devs use to incentivize running the dungeons even when you don't need loot or experience anymore. Granted, FFXIV has many hooks to keep people doing old content (leveling other classes, daily roulette rewards, gil, etc). Pets are just another such tool.

Besides, the pets are less fun to get if everyone has them, right?

Before anyone asks no, I don't have any Heavensward pets other than Unicolt and the various wind ups everyone got for things like early access.