My bank's android app has more control over my account than any other way of interacting with them, including going into branches and posting forms. That's not strictly a monopoly but I'm on my fifth bank and they've all been similarly rubbish.
Huh, interesting. I can't think of many notifications I would want from my bank. I've got an alarm for if my balance goes under a certain threshold, which I guess could be handled via a notification, but IMO it is easier to just make the threshold high enough that I can treat it as a "check-once-per-day" thing, and get the notifications via email.
As a UK customer, I use Monzo bank and have pretty consistently had good experiences. They are a new bank. Recieving international money transfers was the only bit that they didn't support.