I agree with your sentiment that the Fed isn't "out to increase unemployment" but I think it's not entirely correct to ignore the trade off of raising rates. It's the low employment that allows the Fed to keep increasing rates.

The Fed has a dual mandate for a reason which I believe is based on Phillips curve (https://www.investopedia.com/terms/p/phillipscurve.asp) that inflation and unemployment is inversely correlated. Now, it's fair to question this relationship...(I don't think it's as rock solid as it used to be), but the Fed's dual mandate is based on this theory. So, in some sense the goal of lower inflation implies higher unemployment.