You can already tunnel traffic over HTTPS, and I've never been to a coffee shop or library that blocked that. Have you tried that when traveling? I used to do it frequently something like 15 years ago to use SSH, Remote Desktop, anything. (Back then it was a software package called "orenosp" that you set up on your client laptop and then a server of your choice, like your home computer -- not sure what tool is popular now.)

On the other hand, public networks would only have to allow QUIC if regular HTTP(S) fallback ever stopped being supported by most webservers, which... who knows if that would ever happen.