I know this debate is about the US banking system, but I feel the need to point out that in most of the EU, cheques basically aren't used, and business purchases are generally made by wire transfers.
Without a chequebook or credit card, you could offer a bank account without any risk that the bank would ever have to pay out more than its nominal balance. Bank runs could still be a problem, but I feel an appropriate pricing structure could solve that.
Wire transfers can be reversed. It's rare, but when it rains it pours. Maybe this could be "solved" by holding all incoming funds for, say, 60 days but that would be extremely hard on most businesses.
Without a chequebook or credit card, you could offer a bank account without any risk that the bank would ever have to pay out more than its nominal balance. Bank runs could still be a problem, but I feel an appropriate pricing structure could solve that.