The Fed doesn't care about the cost of servicing the debt. That's the US Treasury's job. By law, the Fed has the dual mandate to keep both inflation and unemployment low. That's it. Nothing to do with the cost of servicing the Government debt.
If the interest on the Government debt becomes too high, nobody will point the finger at the Fed. If however inflation is high (like now) or unemployment high, you can start hearing people accusing the Fed of gross negligence. In the extreme, the Chairman of the Fed may be sacked, then brought in front of various Congressional investigations, and may even find himself in contempt, or some other very unpleasant situation.
Bottom line: the Fed really cares about inflation, and doesn't give a damn about debt servicing.