@poyo facts! Yeah the site timer is "wall time", aka it remembers the time you started and shows how long since them. But can't track sleep states which on the switch do pause it.
It's possible that the builtin timer isn't actually a timer, but stops after 400 internet checks (that occur every 3 seconds) or 20 minutely checks, it gives up. More info on those requests is here: https://browsedns.net/topic/25/browser-timeout-this-screen-will-be-closed/34
I have tried faking these responses with different combinations of pretending that the connection works, and adding networking delays to the responses, but everything just still eventually dismisses the browser after 20 min of non-standby time.