I personally only have Protonvpn to test and i'm not able to able to reproduce this problem.
Kagi uses GCP for it's hosting and that tends to have issues with vpn's BUT that doesn't explain why the issue specifically happens when you're visiting a non-kagi site 🤔
The pages currently open in the app itself instead of a browser window (bug right now) I'm wondering if this'll get solved when that gets solved