You are right there.
>>"The Fed wants to keep the price level on a stable trajectory, and they use an interest rate target as an intermediate instrument to do so."
Yes, they use it like the main instrument to do so. And that's the reason why they can't just allow the interest rate to float freely. And because they can't allow the interest rate to float freely they have to add reserves when the inter-bank market is going up. So, reserves will be added to the system when banks need them.
Causality: banks lend -> reserves are added
They can easily choose to let interest rates float freely.
Of course, even if they directed targeted eg the TIPS spread without worrying about interest rates, in the end they would still add reserves when banks lend. (Ceteris paribus.)
(Other indicators the Fed uses include internal forecasts.)
Nobody thinks that interest rates are a good indicator of inflation. Certainly no one at the Fed. Or, what do you mean by 'indicator'?
The Fed wants to keep the price level on a stable trajectory, and they use an interest rate target as an intermediate instrument to do so.
An instrument is a very different concept from an indicator, in my understanding. It's like the accelerator pedal vs the speedometer. Do you use the two terms in a different way?
> But changing the interest rate in the process.
Yes, but as a side effect. Just like the Fed actions also influence the exchange rate or the price of gold, but neither of the two is their target nor instrument.