No, the fed doesn't set rates. Not by much, not for long, not without inflation taking off. In theory they chase R*, in practice they chase the 2-year. Plot EFFR and US02Y on tradingview and tell me who leads whom, and tell me what happened when the fed tried to fight the market.

For the most part, congress decides to spend a certain amount in excess of receipts, congress decides the debt, the private sector bids on the debt, and the marginal bid sets the interest rate.

When the federal reserve steps in they can use their magic balance sheet to, at great expense, tug the interest rate around a little bit. However, artificially creating ZIRP in a non-ZIRP economy would not be a little tug, it would require buying most or all of the unattractive bonds. In 2020 it would have required $25T not $4T (GFDEBTN vs WALCL in 2020). Because the sum total of their intervention was small compared to the debt sold to that point, the low-intervention approximation is correct during the 2010 ZIRP era, the "P" in ZIRP is a misnomer, and "capital had more money than they knew what to do with" is the correct read on how ZIRP happened.