But it's the other way around, if you improve your old device by installing a up to date Android on your vendor-abandoned previously vulnerable device, you go from working banking to banned from banking.
We’re not just talking about the freedom to run software on your own device here, we’re talking about interacting with outside systems. There is an important distinction in context.
As long as it adheres to basic web standards, I believe no, the bank should have no say in what browser you use to access their webpage.