I don't know which banks you are using but in my case I work with five Spanish banks and I can do everything from their websites, no app required. Yes, they try to push you to use their app, some tried to activate mobile 2fa for me when this psd2 thing became mandatory but I always told them their app doesn't work on my phone (which is true) and they offered me alternate methods like sms.
In my country we have a large religious population who eschew the smartphone. This means that no government, banking, or other services require a smartphone.
Can you access their websites without the need to confirm 'who you are' with their app? In my case, not anymore.
My bank used to have other options but it has made mandatory the use of their app.
> Can you access their websites without the need to confirm 'who you are' with their app?
Yes, none of them required me to use the app a single time. In fact, for all the banks I work with, I always identified myself at a local office when opening the account for the first time, the last one less than a year ago. And all of them allow me to operate in the website without the need of an app (actually I could never use any banking app as my telephone lacks Google Play).