This isn't a standards problem, this is an "Apple, Google, and Microsoft made using your own software difficult" problem. They have an interest in making their authenticators the default, and the settings UX on all three platforms makes it unclear that their authenticator is active as well as your own "third-party" authenticator choice.

It’s difficult to disentangle one from another, when the companies with the vested financial interest in locking you in are the board members of the consortium creating the standard.

you can see in both the lastpassword and the bitwarden tutorials that the default iOS password manager can be toggled on or off, just like the other options.