Okay, I figured out a workaround. When launching games, I saw the following lines using strace:
[pid 1957992] connect(13, {sa_family=AF_INET, sin_port=htons(53), sin_addr=inet_addr("127.0.0.53")}, 16) = 0
[pid 1957992] sendmmsg(13, [{msg_hdr={msg_name=NULL, msg_namelen=0, msg_iov=[{iov_base="c\276\1 \0\1\0\0\0\0\0\1\4Core\3lan\0\0\1\0\1\0\0)\4\260\0"..., iov_len=37}], msg_iovlen=1, msg_controllen=0, msg_flags=0}, msg_len=37}, {msg_hdr={msg_name=NULL, msg_namelen=0, msg_iov=[{iov_base="\245\270\1 \0\1\0\0\0\0\0\1\4Core\3lan\0\0\34\0\1\0\0)\4\260\0"..., iov_len=37}], msg_iovlen=1, msg_controllen=0, msg_flags=0}, msg_len=37}], 2, MSG_NOSIGNAL) = 2
For some reason Proton is trying to connect to Core.lan which is just localhost/127.0.0.1 (Core is my hostname, .lan is being added automatically by Fedora I think). This works when the VPN is disabled, but the VPN messes with that somehow.
The workaround is to add 127.0.0.1 Core.lan to /etc/hosts. After doing that, games launch in a couple of seconds again with VPN on.
But why does this even happen -- what's the purpose? And why use the hostname to connect to localhost instead of just using 127.0.0.1?
Nothing extracted yet.
For some reason, all Proton-enabled Steam games take about a minute to launch when my VPN (Mullvad) is turned on (when it's off, launches take a few seconds at the most). I was not able to fix this by changing any VPN settings (like DNS content blockers, IPv6, etc.). Why does Proton need to make any network requests when launching games, and why would a VPN cause problems? Is there any way to disable/work around this?
The only similar report I can find anywhere is here (not resolved): https://www.answeroverflow.com/m/1156372760298266815