are there really countries where the bank doesn't have a website you can use to do a transfer, but you could do it through an app?

I don't know, though certainly the experience is a lot simpler without the 15 minute timeout, painful login, and extra security checks I see on web banking.

Edit: Not to mention that many of the newer banks don't even have web banking. It's app only. Of course, its your choice to open an account there though

In Germany and I think the whole EU 2 factor authentication is mandatory, for which the favored implementation is an app. SMS TAN is out, the alternative is a secondary device you stick your card into.

Do you need a proprietary app for that? TOTP is fine, you can just pick your own.

Haven't seen a bank offering software TOTP in Poland. Over a decade ago, before smartphones became ubiquitous, I've seen a bank offering a physical TOTP device. These days, as far as I've seen, it's either SMS codes or single use codes on a physical scratch cards (haven't seen one in 5 years, though), or in-app confirmation.

Yes and they tend to be shoddily programmed security theater. My bank makes me use SecureGo plus, which goes as far as redirecting you to a website telling you screenshots aren’t allowed when you try to document its latest glitch (which may be another misguided “security” feature, who knows).

No, but there are bank accounts that are app only. Monzo in the UK is a popular example.

In Germany you can use the website BUT you still need the app for 2fa. SMS is no longer an option for most banks, because it is considered insecure. Same goes for TOTP since this can easily be replicated, if you have access to the device generating the TOTP.