Running on Debian 13. After lots of searching, I finally figured out that the problem was with apparmor:
[476956.590593] audit: type=1400 audit(1768920707.057:1017): apparmor="DENIED" operation="open" class="file" profile="snap.chromium.chromium" name="/proc/tty/drivers" pid=245962 comm="ThreadPoolForeg" requested_mask="r" denied_mask="r" fsuid=1000 ouid=0
The only browser I had that is compatible with web.esphome.io was a snap-installed chromium. To fix it, the following worked for me:
snap connect chromium:raw-usb
Not sure where to put this so that others find it more easily than I did.