How often do you have the problem with the mobile site not recognizing you? I just signed on for the first time in a long time, got the code and logged off and on several times. Seemed to be okay. Actually I kind of like the IE mobile version of the website better than the view in the old app. True, it doesn't allow remote check deposits but I never used it any way because I don't get many checks. The Chase site, viewed in mobile allows me to do everything else I do often, like transfer money and pay bills.
I'm a Chase Private Client so it wouldn't make any sense for me to change banks. But I didn't often use the Chase app, and it's been years since I logged on to the main site from my phone (I live and work close to branches and am often at my desktop or have my laptop).
I'd be annoyed and feel differently if I had to get and supply a new code frequently, but I'm finding these days that many places require that even on standard PC accessible sites.