There's a significant concern that Alpha or Delta could be hiding out in an animal reservoir, for example white tail deer.

https://www.pnas.org/doi/10.1073/pnas.2215067120

Anything that emerged would be quite distant from the earlier variants, having evolved in animals for years. But if it still has a human compatible spike protein, that would be bad. Omicron's spike changed drastically in isolated evolution; it wasn't competing against other circulating viruses. Omicron proved very fit.

The primarily barrier to zoonotic disease is that they aren't efficient at going after human receptors. Something that was adapted to humans first is potentially dangerous.