I use it on Android and it's always been a Bitwarden-specific pop-up that shows up on password fields. Maybe with accessibility mode it would work as you mention? Ok that's a new thing to test.
I don't think that the visual indication means they aren't using the system way. The accessibility service is probably not a good idea given how much control it can give the app over your system.
Otherwise I'm not sure how Bitwarden saved passkeys could even work at all without using the native integration.