Well, there are PQC (public key) signature schemes based on hash functions, but they all have large signatures (in the multi-kB range) and other drawbacks (eg being stateful).

Moving to a MAC is also not a crazy idea in a lot of deployments. You can horizontally scale an internal MAC validation service. I believe Facebook use MAC-based tokens internally (CATs).

There are hashing modes that achieve both integrity and authentication. It usually involves prefixing the data with a server-provided secret before hashing. Only someone who knows the secret can produce such a hash, thereby providing both authentication and data integrity.

Cookies are signed by the server, not the client, and only the server needs to check the signature, so there really isn’t any reason to use much more expensive asymmetric cryptography. And post-quantum security is just a bonus.

In the context of cookies, authenticated hashing IS a PQC signature.