Possibly this was hyperbole but in any case it's not correct at all.
Anecdotally, of my two EU (massive legacy French) banks, neither requires a mobile app. SMS all the way.
Even Wise, a cutting-edge neobank, does not require you to use its app. And its website accepts standard TOTP authenticator for 2FA.
Revolut is app-only, which is why I never use it.
No SMS at all (which is not surprising, because SMS is not secure).
Also, IMO fingerprint/face-based authentication is much nicer/quicker, especially for online payment flows like iDEAL (Dutch predecessor to Wero). And banks here work on GrapheneOS, so not much is lost.
Until they don't.
My wording was bad, sorry; but try to install their app just once. After that, I'd bet you won't ever be able to go back to SMS validation (which is what I was talking about at the end of my comment).
If not, I'd be curious to know the banks you're talking about (to consider switching to them, for one thing). What I said above is true of Caisse d'Epargne, HSBC, CCF, among others.
Fortuneo (internet-only subsidiary of Crédit Mutuel) and LCL. I have had both their apps installed at points in the past. In both cases they defaulted back to SMS 2FA upon uninstalling, though I remember worrying I would have the problem you describe.
Ultimately I can't see how a bank could get away with forcing (rather than just pushing) existing customers to install an app. This would surely be a breach of contract.
Can you go in branch and get that fixed?