Late reply, but the app toggle is always on in my device. But when I visit test.nextdns.io on a browser, it may report a NextDNS endpoint or say “unconfigured”. There’s no way for me to know if it’s working or not at any point in time. It’s broken and the app is abandoned (with no updates since 2020). I don’t want a permanent VPN profile on the device since I’d like to have the flexibility to turn it off if/when needed.

Parent might also be getting bit by Apple's iCloud Private Relay and/or the "Hide IP Address" feature of Safari.

Late reply. I don’t have any iCloud+ subscription and thus don’t have iCloud Private Relay. The NextDNS iOS app is broken and abandoned (no updates since 2020).