> the Fed doesn't actually set interest rates
It's not clear cut as to what degree interest rates are exogenous inputs that central banks respond to, but the Fed absolutely does set interest rates, allowing some variability between upper and lower bounds.
Today, Fed adjusted interest on reserve balances (IORB, formerly IOER/IORR) to 40bps from 15bps. This rate determines how much interest banks are paid for reserves kept at the Fed. In theory, this rate acts as a floor for the effective fed funds rate. In practice, it's somewhat murkier.
The Fed also sets the discount rate (now 50bps), which is meant to act as a ceiling on rates. Banks are able to borrow money from the Fed's discount window if they need it; however, there's a stigma associated with utilizing this facility. The Fed now maintains standing repo and reverse repo facilities to help banks manage liquidity.
These policies all target the front end of the yield curve, which is where Fed has the most control. To manipulate the long end of the curve, Fed implemented QE. Other central banks (e.g. BoJ) have gone further, using yield curve control to explicitly impact the term structure.
> Instead they purchase and sell treasuries to member banks, such that those banks' balance sheets change in such a way as to make money more or less expensive to trade amongst themselves, which has knock-on effects for consumers.
Repo rates are determined by the market, but are bounded by the rates at Fed's repo facilities. A catch here is that not all market participants have direct access to these. While repo rates may impact behavior, the Fed's intended mechanism is IORB, which (ignoring steepness of the yield curve) influences how attractive banks find loaning money to clients.
> the mechanism that causes banks to have to borrow from each other (to meet the nightly reserve requirement, historically)
This market used to be the Fed Funds market, which consisted of uncollateralized loans between banks. The fed funds market is basically dead, replaced by the repo market, which is collateralized. IIRC, the remaining participants are GSEs like Fannie Mae and Freddie Mac, which can't collect IORB. They sweep their cash to banks and split the interest (which is why IORB can act as a ceiling instead of a floor).