Average guy will gladly pay a fee to have someone else protect his coins. pretty obvious
I've been thinking about launching a service that is akin to offshore paper wallets. But ... maybe I'd have to be a bank of some sort. Otherwise I just operate on reputation alone. Maybe later. I don't know.
Average guy will pay fee, forget password, lose 2FA key, throw hissy fit, accuse you of stealing when you won't give coins back on the strength of an all caps ungrammatical email.
As part of the service, I will require identification in high resolution, as that is the only way I can verify you, IN PERSON. Which means you have to come fly here to my location and we meet.
If you are trusting me with your coins (probably more than 100 BTC) then you should have no problem trusting me with your IDs.
I'll have a part of the clause that if Tom Cruise shows up at my door step with your credentials (your passport and your driver's license etc) and your face (Mission Impossible), then I won't be held liable. The whole exchange will be recorded in 1080p HD so if the real you suddenly appear, "hey, I thought that guy was you."
I can safely store your cold wallet in a highly secure location. Just not secure enough against the IMF or James Bond or any other highly trained secret agent. (For reals, I have a private vault inside a military base.)
Again, I haven't found any customers, so ...
Banks generally operate on reputation alone (although they also often rely on the reputation of the government insurance that backs deposits as well).
Yeah, there's FDIC for the US and PDIC for the Philippines, but they only cover a certain amount per depositor (and not per account.)