Looks like 70% of cloudflare requests are using post-quantum TLS! https://radar.cloudflare.com/post-quantum
The idea would be to use an existing connection (established via normal TLS) to agree upon and exchange a pre-shared key. For subsequent connections, that key would be incorporated as one extra input to the key derivation function, and a new key would be derived.
This would make TLS more secure against adversaries who can break asymmetric encryption (now or later) and monitor some of your communications, but who do not have access to all of your communications. If you managed to get on an unmonitored network even once (foreign hotel WiFi, coffee shop without a wiretap), and securely establish a PSK there, all future interactions would be inaccessible to an adversary who can't break symmetric crypto.
Probably not too many, because anonymous political speech from 10+ years ago isn't that interesting. Punishing people a decade after the fact isn't very effective for anything.
It sends a very clear message: even if it takes a decade, we will find you and punish you.
The CADO-NFS page gives some benchmark results for 16 threads, suggesting the algorithm parallelises at least somewhat well.