But we already do, she's called Phoebe. Yes she's technically an ostrich, but the same can be said for Cranston and Flora, who are different species deprived in the same Ostrich species category.
I kinda like the BSing they do where they have the default species and then occasionally make them into some other species (especially mythological/fantasy ones). Drago being a Dragon, Julian being a Unicorn, and Phoebe being a phoenix. I think it'd defeat the point to have such a fantastic creature be an entire species. Having a single phoenix is kinda the draw to it, y'know?
I agree with @/michaelsmells, I feel like making them their own species would take away from the uniqueness Phoebe has of being the only phoenix villager. It’s like how if unicorns became a species, Julian would feel less special. Also, phoenixes being a species just doesn’t really make sense… they still have so many real life species left that could be used before making species based on mythical creatures. It would still be a cool species because phoenixes are inherently pretty cool.
Although… if phoenixes did become a species, Nintendo should name one of them Blaze… just because.