I think institutions here could play a key role in the future. There may will be many institutions offering this service somehow, competing with each other, and they will have to prove how trustworthy they are, maybe with algorithms or avoiding hacks/leakages.

Over time we would have some trustworthy new institutions where people could backup a piece of their key.

But I think it's a whole new set of problems, right? What if banks end up playing this role? it would be funny haha.