> Does this mean all American banks (indirectly bank customers) will pay to cover depositor losses that exceed insurance funds?
That assumes banks balance this liability by reducing payments to customers rather than reducing profits. That's a common and completely misleading claim by businesses - if they are taxed or fined, they pass it on to their customers (obviously, it's an attempt to create political support for the business).
The reality is that the ability to raise prices (or lower interest rates on deposits) depends on the elasticity. If you raise prices on your bottle of water at the supermarket, then people will just buy the bottle next to it - the water-maker will be paying and fee or tax increases out of their profits. If you have the only bottle of water in the desert, you can charge whatever you want. I would think that regular savings deposits, at least, are easily moved to another bank.
Another consideration is that if they could squeeze more out of customers, they'd probably already be doing it. By that theory, at least, they've already optimized or that and can't charge more.