A bank is a tool, I don't care how pretty it is, I care how functional it is.
The designs don't necessarily look outdated, they look out dated to us because we live and breathe that stuff.
I'm not speaking in absolutes here though - some bank sites do need to be completely reimagined. Chase specifically comes to mind for example, but Wells Fargo's site was perfectly usable and did not feel dated.
To go even further, I'm not entirely arguing against redesigns, I'm arguing that if you do go that route it should at least be simple and straight forward, it's a bank site not the next social media behemoth.
It's always a really tough problem to keep old users happy who've been using the old sites for possibly years while treating new users to something more modern. This is one of the toughest problems I've encountered working in this field and haven't yet found an elegant solution.
Some of those sites had been more or less unchanged since at least the early 2000's.
A shining example of if it ain't broke, don't fix it.
I'm 99% certain people weren't leaving any bank because of an old website design.
The new ones have so much padding and empty space, and hide everything behind "clever" menus. I'm a PM and avoid this clever bullshit like the plague.