Until now, I have never run "a custom ROM or something", but just the Android that came from the phone vendors and its updates.

Nevertheless, I do not have a Google account and I do not intend to have such an account.

Of course, this means that I cannot install any app from the official Google store, even if it is a free app. The requirement to login into your Google account should have existed only for payments, not for downloading a free app, but nonetheless Google does not work this way.

I already had problems with a bank that has terminated its Web-based online service, replacing it with an app that they refuse to provide for downloading, so that I could install it without having to open a Google account. Therefore I have also terminated my accounts with that bank.

I hope that this behavior will not spread to all remaining banks that still have Web-based online access.

You could try aurora store with anonymous accounts, though that has the problem that other people may be able to see the apps you install.