Voltaire your last paragraph is interesting in the questions. My opinions only, based on having been around this stuff for a while.
Websites becoming too complex. Lots of us here remember the early days when you had at best a dialup serial connection (56k anyone?) and browsing, especially early online shopping was painful. It was obvious the testing was done on the internal 100M network and it worked fine, but flash and active pages with blinky spinny things were painful on dialup.
Heck most current residential broadband is more than commercial businesses used to have.
Cookies: interesting when the cookies start leading to all the third party advertisement stuff, think Google ad services that also try and triangulate location to give "a more tailored user advertising experience" when we are all going "No, disable that, turn this off, AdBlocker this"
Mobile users. A smartphone is a different beast compared to a desktop. Tablets are basically smartphones that you can't stick in your pocket. Websites written for a desktop, don't translate directly to a mobile device, I typically avoid using a browser on my phone (I hate having to scroll in 43 dimensions, tapping 87 times to expand text so I can read it, etc), so how does a business make the same thing easily accessible to the multiple platforms, in a "cost effective" manner? I don't know the answer to that; I'm not a web developer, but I think there are supposed to be some extensions to standards to a server can identify the user is on mobile and should do soemthing different.
As for the OP and his bank; I have no answers for that.