Are there any mainstream websites which only allow a single TOTP token to be enrolled? I can't remember having ever run into that issue. I do recall it occasionally being an issue with Passkeys, though.
The non-disclosure is indeed neat, but the same can be achieved with a password. For example: generate public/private keypair on account creation. Encrypt private key with user password. Store both on server. On auth, client downloads encrypted priv key, decrypts it with user-entered password, then signs nonce and provides it to server as proof of knowledge of user password.
I would say the majority of services I have TOTP set up for only support one token at a time. It's only the bigger, techier services that have support for multiple.
You don't need to involve a private key there. Modern password authentication algorithms never reveal the bare secret (outside of initial registration ofc). For example, PAKE uses Diffie-Hellman coupled with the (salted) password hash to independently derive the same session key on both sides of the connection.
AFAIK the primary technical concerns are insecure storage by the server (bad hash or salt) or keylogging of the client device. But the real issue is the human factor - ie phishing. As long as the shared secret can't be phished it solves the vast majority of real world problems.
Point being, TOTP on a rooted phone handled by a FOSS password manager app whose secret store the end user retains full access to will successfully prevent the vast majority of real world attacks. You probably shouldn't use a FOSS password manager on a rooted device for your self hosted crypto wallet though.