There is one flaw in your theory. Custodial assets are not part of the fractional reserve system and thus, not to be lend against.

Custodial assets are held for safe keeping (for a fee of course) and, in most instances, they custodian does not have access to the funds. There are times when the custodian is given the power of attorney to handle certain obligations within the account, for example, the custodian is operating on behalf of a minor.

That said, not your keys not your bitcoin takes on new meaning in the banks. Anyone who puts their bitcoin in a bank obviously has no idea what is going on.

As for the paper of Bitcoin, we already saw that with the future market and other derivatives that are out there. Of course, they arent true bitcoin but people are trading in them anyway.