Sure the Fed isn’t entirely private and really you could chalk it up to a public/private partnership if you want to label it.
If it was a government tool like you say then how they were so late with rate hikes? They ignore government officials both appointed and elected who called for an increase of rates and now a decrease. I mean they would work to help the government not cause pain and thus result in people losing trust.
You can only chalk it up to a public/private partnership at best