MAC where key is provided afterwards
If you have a secure MAC algorithm $M$ and a collision-resistant hash function $H$, then you can define another secure MAC algorithm $M'$ such that $M'(x) = M(H(x))$. Historically, though, HMAC was defined as a construction that provides reasonable security as a MAC even for an underlying hash function which is a bit flaky in that respect (in that case, Merkle-Damgård functions like SHA-1, that suffer from the so-called length extension attack, which illustrates the subtleties of MAC-building out of hash functions).

(For reference, my comment in the TLS-1.3 mailing-list is about the need to buffer incoming data until the MAC can be verified; this is unrelated to the moment the key is injected in the MAC process.)