You probably can't use a startup bank, as the app is one way they cut costs. Use a boring bank.
Personally I really hate that they want to kill the card, and go to an app model (with the option of a key fob). It's made even worse by the police (and others) pushing the idea that the app is safer. How is it safer to have your two-factor authentication app on the same devices as the apps that need the second factor? Those things should be separate.
Barclays don't require the app. You can still use the website or even telephone banking with a real agent. They do give you the chip and pin reader, the same as the one used in branches which works well.
Santander do seem to use a mobile for 2FA but only to send a text message and you can still use the website.
I rarely have to do more than 1 banking transaction / week so I just use the website, from home. Not too bothered about having another app, especially with access to my bank account.
I don't know about Ireland, but in Portugal most banks have a Web app and not just a mobile app.
