> If everyone stashed money in mattresses (as if), then all the banks would be forced to start enticing would be depositors with “store your money with us we will pay N% interest!”. This would bring out the savings from mattresses to be used.
No, banks would not be forced to do that. First, banks don't need deposits and some big banks - investment banks - refuse to even accept deposits. Banks make money by borrowing from money markets at one rate and then lending at a higher rate. Their value is credit analysis and credit intermediation.
E.g. if you are a household, then you would like to borrow for a mortgage. Well, the Smith Family does not have access to the bond markets. But Big Bank does. Big Bank makes loans to thousands of families like the Smith family, it does all the credit analysis checking their work history, etc, and then it either sells bonds to raise the money for the mortgages, keeping the mortgage on its own books, or it packages up the mortgages and sells them off as mortgage backed securities.
It is the fact that you can't borrow from the money markets but banks can that explain the existence and profitability of banks, as they sit between you and the capital markets. If you kept your money under a mattress, the banks would still sit between you and the capital markets. If everyone emptied their bank accounts overnight, the bank would need to sell some assets for reserves, and then buy paper money with the reserves, and then give you the paper money. The Central Bank would then purchase assets creating enough replacement reserves and the system would continue as before, just with a lot of money stuffed in mattresses. In fact this isn't even theoretical, as there is a lot of drug money in the form of cash hiding in Latin America. It's not in mattresses but it has left the U.S. economy and no one cares as we just print more.
Whether or not banks want to get into the deposit game is a function of whether they think the deposit fees and customer relationships created will be of sufficient value to offset the costs of providing those services. For many banks, getting depositors establishes a relationship in which you will turn to that bank when you need a loan, etc.
Moreover the interest charged to banks is not subject to manipulation by households that refuse to deposit their money in banks, first because banks don't lend deposits, but second because the bank's cost of funding is determined by the Central Bank's overnight interest rate plus a premium for the credit-worthiness of the bank itself. Whether the bank does or does not provide deposit services is irrelevant. Deposit services are purely sales channels and fee generation channels.