In my opinion, they're too big to fail. As are all the main UK banks. Even Northern Rock was too big to fail, because of the massive loss of confidence it created. And they were mostly a savings/mortgage bank, with many fewer current accounts.
The EU competition action had nothing to do with that anyway, they did it because of state aid. Too-big-to-fail is a job for the banking regulators, not the competition ones.
It's the complication of resolving the bankruptcy while a few million customers lose access to their savings and current account - and all the customers of other banks try to withdraw a few thousand in cash, so they don't end up in the same boat. That's what destroys economies and creates depressions. It's what the Eurozone deliberately did to Greece, and what made the 1930s Depression so awful. Well that and lack of social safety nets.
Technically all retail banks must have a plan to resolve them. I just don't believe any government would dare to use it. It's like the Eurozone. The Italian banking system has been on the brink of disaster since about 2011. The government are unable to borrow more money to bail them out, and the Eurozone have changed their rules so the only way to do it now is for savers over €100k to take losses and junior creditors to be bailed-in. That is so confidence destroying that nobody dares do it, unless at gunpoint like Greece and Cyprus. Worse the Italians and Spanish had a mis-selling scandal, where their banks were selling unsecured junior bank bonds to retail customers, as if it were as safe as a savings accounts! Even though the savings account is guaranteed up to €100k and these aren't.
I think moral hazard is a pipe-dream. There's moral hazard. If we're going to have banks, they must be able to be bailed out by the state temporarily, or we get bank runs.
However we've now ring-fenced the money in retail banking, which means that the bit of the bank you allow to fail is the rest of it. While keeping the retail bank alive.
The other thing we should do is probably to give bankers more of their bonus in shares, that they can't sell for years. So they have a financial interest in the health and survival of their banks. Sadly the EU had the bonus cap, done purely for reasons of populism, even though it's led to bankers getting paid more for bad performance. Yes people take risks for bonuses, but the system also allows banks to cut costs easily in bad times - and if we could make those bonuses into long-duration share options, that would probably also help.