I think you're talking past each other.
A central bank doesn't directly set interest rates for your mortgage.
It can set rates at which it will lend to other banks, which in turn influences the rates banks will offer to mortgage borrowers, but this isn't necessarily so, see for example 2008.
Of course there are more contracts directly tied to the central bank rates, but thats just formalising the thing thats supposed to happen anyway.
Yes. Btw, I'm not sure how many contracts are directly tied to whatever rate the central bank announces these days? It used to be more common to tie contracts to eg LIBOR, which is or was a reported interbank lending rate, not a decreed one.