The Fed can only 'set' the Overnight Rate by being ready to borrow or lend at the Overnight Rate. The Fed still has to stand ready to actually engage in these transactions.
As a counterfactual: we can imagine a world where the Fed 'sets' these rates by just announcing the rate but not engaging in any transactions and legislators pass a law that forbids banks from borrowing from one another at any other rates.
The interbank lending market would just not clear in that case, and you'd either have a glut of demand or a glut of supply.
The Fed sidesteps this by actually borrowing and lending in the interbank lending market. And that's economically equivalent to buying and selling very short term bonds, even if the legal form is different.
As a counterfactual: we can imagine a world where the Fed 'sets' these rates by just announcing the rate but not engaging in any transactions and legislators pass a law that forbids banks from borrowing from one another at any other rates.
The interbank lending market would just not clear in that case, and you'd either have a glut of demand or a glut of supply.
The Fed sidesteps this by actually borrowing and lending in the interbank lending market. And that's economically equivalent to buying and selling very short term bonds, even if the legal form is different.