You're actually onto something, I think. Half of the fed's job is setting interest rates, but the other half is setting interest rate _expectations_. If people expect rates to hold steady or decrease, they may not change their behavior (e.g. slow hiring or spending) to the extent needed to reduce inflation. But if they do expect rates to stay high or go higher, then they will act on those expectations, and the effects will ripple through the economy.

And yes, it is by and large "rich people" the fed needs to convince, specifically business and capital owners.