Unfortunately influenza has animal reservoirs that COVID-19 does not seem to. It is unlikely we could permanently eliminate it. Maybe a few seasons of reduced numbers, but that's probably the best you could expect.
Possibly dumb question: since SARS-CoV-2 came from bats originally, are bats not an animal reservoir for it?
They are, they are also a reservoir of many other unknown diseases yet to come. But people don't get into close contact with bats as often (unfortunately this time it probably happened from the unsanitary Chinese wet markets).
It did not come directly from bats to humans, there was another intermediate host species. We don't know which, pangolins possibly, or minks or some other species, probably in closer contact to humans than bats.
Right. The other thing is that (as far as I know) there are no species in which sars-cov-2 is endemic. It may be that (besides humans) there had only ever been one other carrier of this particular virus. There are probably close relatives of sars-cov-2 that are common in bat species, but not necessarily the exact same strain.