The money they lend out is backed by Treasury Securities.
Part of a president's success is dependent on having these people loan out money wisely such that the government breaks even (or even makes a little profit) in lending out the money and doesn't get stuck with too many delinquent loans.
These folks could also indirectly set the reserve ratio by either only lending to banks with a reserve ratio above a certain amount, or charging more to banks with lower reserve ratios.
Banks would be regulated to some extent by some other agency. That other agency might provide a grade to each bank in terms of their past performance and current financial state. This grade could also influence the rate at which these banks could borrow money.
Also, these banks would also be able to loan money back to the government to help the government cover other bank loans without having to issue more Treasury securities. Sort of how some businesses/people can sell power back to the power companies if they are a net generator.
In this way the Fed (or whatever we call it) is more a passive loaner of last resort rather than a mover-and-shaker in the world of finance.
It sounds fairly logical, but the world has a way of exceeding or transcending mere logic. You're for sure hinting at some of the macro-intractions that go on, but hard experience has a way of outwitting some best-laid plans. It's very ironic that I find myself defending the money-creation power of the Fed, but it's IMO clearly necessary in some cases. And yet it can clearly be taken to extremes (and in the last five years this has been done, in spades) For example. Hurricane Andrew blows thru Florida and 3 million people suddenly a: have no way to make a living, b: have no place to live, c: cannot make their existing mortgage pmts. Some of the losses are paid for by insurances of various types and personal savings, but...should these folks have to wait until McDonalds, then Home Depot, then some restaurants, then some mortgage cos all decide to go back into business in the general area...everyone starts as if they are teenagers with minimum wage jobs, living in tents, then some work their way to being managers, then some entrepreneurs start some other businesses, indeed, a whole economy gets rebuilt only from savings? I mean, that could take 10 or 15 or 20 years.
Secondly, what about bad debt? I mean if someone simply cannot pay back their debt, let's ignore fault, should the entire financial system be made to suffer the shortage of funds that result? If funds for lending ran dry, then rates would have to rise, perhaps dramatically, reflecting the scarcity of funds. Rates rising so much would then attract investment in Treasury debt, and maybe the attractiveness of investing in that debt would preclude those funds starting or continuing actual productive businesses. Perhaps taxes would have to rise drastically to resupply the lending pool. Again, the participants of the system would seem to be punished for the bad debts of a few. I dunno, maybe it should be that way. If those rates rose dramatically, probably the stock market would decline and there would be a much wider "impoverishment" than just the poor schnooks who couldn't pay back their debts AS WELL AS the banks that made the bad loans.
In the mystery room you propose with the guys lending out money depending upon supply and demand, believe me, that is not that far from the way it actually works! And that system isn't that bad. You're saying the the pool of funds is (or might be) backed by Treas securities....but...what are THOSE backed by? The short answer is, the govt's taxing ability. Odd indeed is the fact the Fed was born in 1913...the same year as the Fed income tax!
What's more, this doesn't all happen in a US-centric bowl. Foreigners and foreign banks are involved as well. Japan as I'm sure you know has had a ZIRP zero interest rate policy for many years, and this so-called "carry-trade" has had a large impact on worldwide liquidity. Now...I think Japan is going to raise it's rates to just under .1%! Can you imagine?
Anyway, it's a tangled web, and what's more, what the Fed is, or does, or is supposed to do, has taken on the philosophies of its' leaders under stressful times: Volcker in the 80's and Greenspan in the 90's.