It is rare today that banks hold mortgages and use depositor funds to fund them. Most (government backed, such as FHA, Fannie, and Freddie) mortgages are securitized and sold on the secondary market to bondholders (there are caveats for small banks, credit unions, and non-qm lenders).

A narrow bank does not require lending to service deposit accounts. It doesn't even need to pay interest to account holders. It simply needs to accept, store, and distribute fiat on the behalf of its customers via ACH, debit, and credit processing networks. Think prepaid card with no fees and no overdraft capability.

The Federal Reserve is culpable in that it will not approve banking licenses for narrow banks [1], as it challenges the business model of existing entrenched banking organizations. Can you build a narrow bank without getting the Fed to sign off? That is a conversation I would be interested in having over a cup of coffee.

[1] https://www.spglobal.com/marketintelligence/en/news-insights...