It's also a way they can secure their loans. It's the exact same reason Capital One has bank accounts along with their credit card business (and Amex doesn't). Getting credit on the open market is expensive, by getting you to sign up with your bank account, they can leverage it to get cheaper loans for buying stocks (i.e. margin accounts). Capital One targets lower income people, so it's riskier and more vulnerable to fluctuations in market conditions. Amex in contrast, targets higher income individuals so fluctuations impact then less. This they can get away not holding bank accounts and securing credit elsewhere.

This means if there's a downturn in the market, they won't go bankrupt due to all the outstanding margin accounts (or have to do margin calls).