Is Cloudflare’s proposed ODOH protocol anything that could improve Bitcoin’s privacy?

Bitcoin is running with onlynet=onion in seriously unsure, no one should ever be running the software in this way. There is no sybil protection, it is perfectly feasible to execute an unlimited number of HS peers and prevent a node from connecting to one that will give them a true vision of network consensus. There is no cost to carry out this attack at all.

one must either leave DNS queries enabled (which could possibly reveal that a user is running Bitcoin)

DNS queries can also be delegated.

Does (theoretically) using the ODOH protocol for initial node discovery make any sense from the privacy perspective?

I suspect it makes any meaningful difference, especially since seeding is just something that happens for a goal starter and likely never again. You are almost never going to be able to hide that you are running a Bitcoin node from your ISP, even by using a proxy the traffic patterns are obvious in Bitcoin.