But I also think this confuses topics. You are just talking straight up arbitrage that has nothing to do with being a bank. That arbitrage from high supply area to low suppy area will lower the price. The arbitrager makes some money on the transactions, but the net effect is to push the price down. Not that great for the huge socks on loans already on the banks balance sheets.
While the way the Fed operates, currency enters our economy through bank deposits turned into loans (not as true as it once was in the post QE world) those loans are going to be repaid 5+ years when inflationary effects have already started pushing the dollars down that they are being paid back in. Plus, the point of the extra reserves is to drive down rates since there will be a greater supply. They get hosed in both ways.
Moreover, being a bank is about arbitrage. Traditionally, banks arbitrage between short term loans (the federal funds rate and the overnight rates) and long term debt. i.e Banks borrow short-term and lend long-term. The way they make money is taking on risk: converting a lot of risk (long-term loans) into short-term risk (the banks borrowing at the overnight rate). This is where quants and models come in. It takes a lot of skill to turn that long-term risk into short-term risk.
Banks do the same thing. Inflation is like a cone of molasses that is being pored on the banks. It then slowly expands. So banks get the cheap money first and lend it to people that haven’t been hit with inflation yet.
If they do this fast enough, they make money off arbitraging the inflation rates. Of course, the CPI doesn’t capture any of this.