I think you might be surprised how much risk Stripe is willing to take on when they can gain hundreds of basis points on cutting out the networks. Only time will tell though :)

Not if it means having to turn into a bank. The repercussions on their business will be draconian. Although I would be surprised if they haven't already explored or fully gone through a money transmitter license for a subsidiary.

Can't they just buy a Bank?

Almost certainly they would, and there might be some small banks out there they could buy or merge with. That would get them out of a lot of legal issues, and also buy them some relevant legal expertise.

However, I seem to recall WalMart investigating this option a while back, and it never happened. It may be that the more you look at the option of becoming a bank, the more it looks like it would take over your life/business. Also, there are doubtless many lobbyists who the existing banks employ who would try to stop it.

I think the issue is being a bank vs owning a bank.

What's the risk if payments were to be debited from the customer's bank directly?

Related side note: I've always wondered, if people don't ever need the "credit" in the credit card transaction, could they get a discount of 2 odd % on each transaction that the CC companies charge, say if they connect their bank accounts directly to they payment methods? This would effectively a debit card but not using the VISA/MC networks. What's stopping Apple/Google pay from processing payments directly from my bank?

Almost none. PIN/passwords render fraud to negligible amounts. That's why debit card transaction have fees for ~$0.10 under Durbin. Companies like Stripe make a killing whenever their customers use debit cards instead of credit cards because they're charging 2.9 +0.30 with a cost of 0.10.

But if you had to rely on debit cards only, then commerce at least in the US would grind to a halt. I believe the average American CC debt is ~$8000.