A two-part question: In the 1980’s, banks once were permitted to sell stocks directly to the public - competing with discount brokers, etc. Then, it was stopped because…? So if you can, please explain why banks don’t sell stock today, but there are websites through which one can???
It’s illogical, Captain! 
(with one eyebrow raised!)
AFAIK, banks weren’t allowed to sell stock directly, though they could set up their own brokerages as subsidiariaries and sometimes set up a desk in the lobby to do this. I suspect they stopped because they weren’t making any money at it.
Banks can’t sell stock because they’re forbidden to by law. That’s because back in the 1920s, banks not only sold stock, but they invested the money entrusted to them directly in stocks. When the market crashed, banks were wiped out. It’s one thing if you speculate in the stock market; it’s another if a bank does.