It's probably the stone-age of PQC right now. Apart from the humongous sizes the amount of scrutiny and standardization is low compared to RSA or even elliptic curve cryptography. So chances are that at least one popular algorithm needs to be swapped out in foreseeable time.

On the other hand speaking about HTTP newer versions allow for more efficient connection reuse and multiplexing so the bulky handshakes aren't needed as often.