Earlier quoted context omitted.
Yes, well... they make their own predictions, so it's not really a matter of whether the public trusts the Fed as long as they are internally unified. Public inflation expectations certainly do play a role in inflation, but, as Volcker showed us in the 80s, it doesn't matter how much credibility you have with the public if you're willing to slam on the brakes hard enough.
But I think a lot of the Fed's strategy was jawboning the rate increases - aka offering "forward guidance". Notice how the rate hikes were fairly measured and tepid. They didn't even start the actual QT until a few months ago. There's a limit to how effective jawboning is, and it requires the public have credibility in the Fed to do so. Eventually, you have to back it up with actions.
If they were truly going for shock & awe, it would have been much more effective do one single 2% hike rather than breaking it up over 3 adjustments.