I have seen a type of banking proposal where the bank always borrows from the central bank at x% and issues loans that way and just passes the x% cost to anyone holding the money. That means no deposits were required to begin with. If anything, cash means that the person who withdrew money would be stuck with the borrowing fee until they deposit their cash back in.

So yeah although this is still theoretical, it is indeed possible to do banking without having a single customer cash deposit.

In that proposal, it seems like all the risk of the loan is carried by the bank with none of the profit. How would such a bank be profitable? What happens if the loan is not paid back?

Parent meant x%+y% or some other scheme for risk management and profit.

Well, bank lending creates deposits, so they need to deal with deposits to actually be a bank.

There are non-bank lenders (and non-bank payment services companies that offer prepaid visa/mastercard products), but they ultimately are then customers of the bank and they are much more limited in what they can do that banks are. That's why it's crazy difficult to get a banking license.