Is there any part of the lore that says that Phoenixes don't exist? Because that's what really matters. Just because there's no lore that says phoenixes exist doesn't mean they don't.
I dont think there is any that says they do not.
I also dont think it directly contradicts lore, but a phoenix' lifespan is cyclical, and TES creation is a downward spiral.
Maybe a spirit that escaped dissolution into the earthbones by realising it could have a cyclical existence, instead of withering and dissolving?
Are you asking for fanfiction reasons?