They're equivalent to big banks, and big banks never die. Even governments bail them out when they screw up.
Bottom line, the bankers always wins. That's how the world is setup today.
I'm not a financial expert so this is just my naive view of all the crap that goes on in the financial world. Hope I'm wrong...
Big banks screw up impressively, but not consistently. They can go broke and survive, they can commit massive fraud and survive, but they can't be utterly useless at the very act of banking and survive.
Paypal is rapidly approaching that point. It's name is almost synonymous with "we'll ban you without warning, and freeze your funds when we do". It's literally the first thing I expect when I see 'Paypal' in a headline.
Big banks survive partly due to FDIC insurance and similar - they're fairly unlikely to simply strip all the cash from depositors. Paypal, by contrast, has ruined everyone from Amazon sellers, to Kickstarter campaigns, to startups. At this point, I can't imagine starting a business that relied on Paypal without having another, more reliable way of handling payments.
Put it this way: I would trust Chase to handle my paychecks, even if they do make bad investments. If they started randomly freezing 2% of deposits, I would sure as hell not continue trusting them.