Is the Cloudflare proposed ODOH protocol a thing that could improve Bitcoin node privacy? Recently Cloudflare proposed an improvement to the DNS over HTTPS protocol called ODOH (Oblivious DNS over HTTPS). It allows separating a DNS server into two parts: the resolver and the proxy that receives DNS requests from an end-user, thus avoiding leaking an IP address to the resolver and an actual query to the proxy server.
When running the Bitcoin node with the onlynet=onion parameter, one has to either leave DNS queries enabled (which could potentially reveal a user is running Bitcoin), either to disable the DNS queries completely (dnsseed=0 and dns=0), and then to seed some good nodes with the addnode= parameter.  Both options aren't perfect.
Does (hypothetically) using the ODOH protocol for the initial node discovery make any sense from the privacy perspective?
Bitcoin running with onlynet=onion is severely insecure, nobody should ever be running the software in this mode. There is no sybil protection, it is completely feasible to operate an unlimited number of HS peers and prevent a node from ever connecting to one which will give them a true vision of network consensus. There is no cost to performing this attack whatsoever.

one has to either leave DNS queries enabled (which could potentially reveal a user is running Bitcoin)

DNS queries can be proxied as well.

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

I doubt it makes any meaningful difference, especially as seeding is typically only something which happens for initial startup of a node and likely never again. You're almost never going to be able to hide that you're running a Bitcoin node from your ISP, even by using a proxy the traffic patterns are very obviously Bitcoin.