The Fed as we have seen definitely "has the power" to raise rates. The question behind your question is, to what extent does raising rates lower inflation? I'm not an economist, just some guy on HN, but I'm pretty persuaded on the idea that inflation is, as others have said, a "monetary phenomenon." If you strip away money for a second, the "actual economy" is just supply (people making stuff), and demand (people buying stuff). Now imagine we're back in 2019 and the economy exists with some baseline amount of money in circulation. Then in 2020 and onwards, more money is printed to allow for emergency government spending in the face of the pandemic (a policy, by the way, that I think was completely correct in context). With interests rates low/zero, the money that was added to the economy basically stayed in the economy, bouncing around according to supply and demand. However, once people got the new money, and people consequently decided to buy more stuff with that money, the economy did not magically start making way more stuff right away to match. Classically, this is where the inflation supposedly happened.
As we discovered with some recent bank failures, it is somewhat more complicated than this. People did not just use the new money to buy stuff. They also used it to invest, and a lot of those investments would not have made sense if interest rates were higher. Somewhere I saw the example of an "AI dog-washing startup"; this fake business illustrates the type of real but not necessarily sound business that was suddenly getting investment because there was a lot of money flying around inside the economy. Now, when the Fed "raised interest rates," what actually happened was it created new investment vehicles (e.g. treasury bonds) that offered returns on investment that were much more attractive to investors than the previous generation that offered low/zero interest. Banks and others shifted towards these new, better investments and tried to sell their old, worse ones. Hence, some of the money that was flying around began to exit the economy and return to the government coffers. This was bad for banks like SVB that had a lot of "interest rate risk." It was also bad for downstream investments like the AI dog-washing startup, which were now competing with "better" businesses in an environment with less money flying around—this is where you see e.g. the current tech hiring downturn and layoffs.
So that's about what the Fed has been able to do. One assumption you've highlighted here is, to what extent is the Fed doing all this based on research and deliberation? I think the short answer is, we don't know. Interest rates are indeed a blunt instrument, but they are also the instrument the Fed can control, and so that's what the Fed is using. This leaves a lot of room for conspiracy theories and speculation. I happen to think the Fed is doing its best within the constraints of its powers, but the Fed cannot singlehandedly "fix the economy." They can print money and adjust interest rates. And doing these things affects the economy in theoretically well-understood ways.
Ultimately, raising rates is like putting an ice pack on an injury. It reduces the swelling, which is helpful, but it doesn't fix the injury per se—the fixing happens in an entirely different, more complex system, really a system of systems. Just so with "the economy." The economy ultimately exists as a sort of distributed phenomenon in the thoughts and actions of all its participants. These thoughts and actions are not aligned, and so we get the infinite omni-directional tug-of-war known as the "invisible hand of the market." The Fed certainly has a lot of ways to influence the economy, but it cannot force people to think or act in precise, coordinated ways.