Why is it so crucial to their business model that the customers' deposits are deposited directly with the Fed, rather than lent out to other banks on the overnight Fed Funds market? The Fed Funds rate is typically lower than the IOER rate paid on deposits with the Fed, but only by a small amount:

since early 2009 the fed funds rate has generally been 5 to 20 basis points (one basis point is equal to 0.01 percentage points) lower than the IOER [1]

For example, money lent out on the Fed Funds market on thursday earned 1.92% [2], while money deposited with the Fed earned 1.95% [3].

The Not-Quite-So-Narrow Bank wouldn't be able to pay as high a rate of interest to depositors as the true Narrow Bank, but it also wouldn't be dependent on the cooperation of the Fed, and Congress's authorization to the Fed, in collecting the IOER rate.

While looking up those numbers, i came across a nice detailed series of posts by George Selgin on IOER [4] and the Narrow Bank in particular [5] [6] - well worth a read.

[1] https://www.stlouisfed.org/publications/regional-economist/a...

[2] https://apps.newyorkfed.org/markets/autorates/fed%20funds

[3] https://www.federalreserve.gov/monetarypolicy/reqresbalances...

[4] https://www.alt-m.org/2017/06/01/ioer-and-banks-demand-for-r...

[5] https://www.alt-m.org/2018/09/10/the-skinny-on-the-narrow-ba...

[6] https://www.alt-m.org/2018/09/14/the-narrow-bank-a-follow-up...