I have only ever used one bank (Wells Fargo) my entire life, so I am not a good candidate to suggest banks. I am satisfied with Wells Fargo though, for what it's worth. I can log in through the browser.
I don't know about Ireland, but in Portugal most banks have a Web app and not just a mobile app.
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.
I was confused by the recent change where online purchases require an SMS or the app, and no longer work with the code card.
I think the app is probably encouraged since it protects against a phishing site like this: hxxp://tilbagebetaling1060dkk.net/1060DKK/ -- but I'm not quite sure how.
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.