If any kind of proof about serious quantum computers comes to light, browsers can force most websites' hand by marking non-PQ ciphers as insecure.

Maybe it'll require TLS 1.4/QUIC 2, with no changes but the cipher specifications, but it can happen in two or three years. Certificates themselves don't last longer than a year anyway. Corporations running ancient software that doesn't support PQ TLS will have the same configuration options to ignore the security warnings already present for TLS 1.0/plain HTTP connections.

The biggest problem I can imagine is devices talking to the internet no longer receiving firmware updates. If the web host switches protocols, the old clients will start dying off en masses.

No need for a TLS 1.4.

Leaf certificates don't last long, but root CAs do. An attacker can just mint new certs from a broken root key.

Hopefully many devices can be upgraded to PQ security with a firmware update. Worse than not receiving updates, is receiving malicious firmware updates, which you can't really prevent without upgrading to something safe first.

There is no reason to not support non quantum safe algorithms for foreseeable future in the first place