It doesn’t matter whether it’s a private or public service if they both use the same auth provider (beId). I wouldn’t be surprised if the SMS/TOTP options went away completely at some point for our “security”.
A different issue is that itsme is often the only option when doing things on mobile. Sure, you can avoid it for now, but it’s getting increasingly inconvenient to do so, unfortunately. I try to express my disappointment to itsme every now and then about the fact that they require Google’s SafetyNet and that the Connective Plugin needed to activate itsme in the first place doesn’t even work on Linux, but to no avail. They sent me a detailed email about setting up a Windows VM to get it working so credit where it’s due for the effort, but the situation is still bad…
That’s what I was saying — sometimes the TOTP option is not available at all for certain services. Ente, Aegis, Bitwarden, etc won’t save you there and itsme remains the only option on mobile when that’s the case.