Fractional Reserve Banking and Lone Banking

I have heard the claim that fractional reserve banking is not a problem with a lone banker, i.e., if there is only one bank in which all people have all their accounts.

If there is only one bank, however, I would think that the problem, in a sense, can get worse than it otherwise would have. Because if one single bank can just move around numbers amongst internal accounts, then there would be much less an incentive for the bank to restrain fractional reserve banking practices. “No one will ever come and claim their money, so we can just multiply the money supply, and prices, as a consequence, will go through the roof.” (If we assume that all transactions go through this bank, and there are no cash transactions, this could get really bad.)

As far as I understand, this is one the major causes of central banking being a problem. One of the major roles of a central bank is being the “bank of the banks.” When you have a central bank, and there is a net payment from bank A to bank B, a normal way of making this payment (as I understand) is to move funds from bank A’s account with the central bank til bank B’s account with the central bank. Would not doing it this way reduce the need for having available reserves? So the problem you get with a lone banker is to a great extent what you get with a central bank?

Appreciate your thoughts.

Yes, definitely not a problem for the banker.

Yes.