It does have some challenges in introducing a read-before-write to fetch the session key at the start of the session, but given the relatively low call volume of such flows that might be a small price to pay to simplify security audits and de-risk changes to any service in the call chain.
Unfortunately my understanding is that it’s trivial to implement unsoundly but it’s also not something for which there are an abundance of good implementations across languages.
It’s been awhile since I’ve looked though so maybe there is a newer, less radioactive approach. But yes, never actually sending the authenticator itself (and doing so in a way that the proof is valid only once) would stop this sort of thing cold.