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Privacy DNS Chooser Script v1.0 "Snow Breeze"
(kbin.social)
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Community icon by Alpár-Etele Méder, licensed under CC BY 3.0
Have you looked into how existing software handles captive portals. I believe, both Ubuntu (or Gnome or Network-Manager) and Firefox do check for such portals and detect real internet access. (They simple poll some URL http://detectportal.vendor.com and check for the expected return code. Portals usually redirect.)
Now I'm thinking, what if this check could trigger a change to the DNS configuration. That is use DoT when internet is available, otherwise fall back to DHCP announced DNS
That is neat! It is a specific response so it should work.
This should work. What would be needed is to track the process of the login and only continue when the window is closed again.