The Fed supplies bank reserves. Mechanically, it goes into the "repo" market, which is the market for loans collateralized by Treasury securities. When the Fed wants to expand the money supply, it goes long in the repo market (meaning that it makes more loans), which lowers the interest rate in the repo market. So as a practical matter, the interest rate goes down whether Bryan adjusts his money demand or not.
The repo market, unlike Bryan, gets very exercised over tiny changes in interest rates.
Unless I misunderstand Arnold, this conflates interest-elasticity of money demand with interest-elasticity of loan demand. People adjust their borrowing to interest rates all the time. But cash balances? That's hard to believe.
So is "Bryan's personal money demand function the ultimate determinant of interest elasticity"? Of course not. I'm open to the possibility that my interest-elasticity of money demand is strangely low.
But I don't think it is. When I've asked economists "Do you hold less money when interest rates rise?," they usually admit that they don't. And we're trained to think we should! If we asked non-economists the same question, I bet that most would simply be baffled.
Institional money demand is probably a little more sensitive than individual money demand, but I still doubt the total effect is large.
Economists have come up with a lot of crazy theories about what responds to interest rates. My favorite is that higher interest rates increase fertility, because parents foresee that they'll be able to afford larger bequests. The view that higher interest rates reduce cash holdings is a lot more plausible, but that's not saying much.