Entangled with is a better way to think about it. The Fed strongly influences the rate curve. But the rate curve also influences the Fed. And trading rates isn't just about predicting what the Fed will do (though that's a big piece of it).
How much does Fed actually consider the shape of the yield curve? Powell's recent comments indicated they focus on front of curve, as opposed to something like 2s10s. Aren't central banks like ECB/BoJ more attentive to overall shape?
Quite a bit. The February MPR cites "yields...across maturities" in the first sentence under financial conditions [1]. (One of the entire novelties of OG QE was the purchase of longer-dated securities.)
[1] https://www.federalreserve.gov/monetarypolicy/2022-02-mpr-su...
The Fed frames its policy response in terms of the somewhat nebulous natural rate, which implies that there is some exogenous component to rate policy. Additionally, rates have consistently trended downwards for 500 years, well before the invention of central banks [0], supporting the notion that rates are driven by exogenous factors.
[0] https://econreview.berkeley.edu/the-long-decline-of-global-i...