I am reading Goldreich, Vadhan, Wigderson: Simplified Derandomization of BPP Using a Hitting Set Generator and trying to understand the result that polytime hitting set generators (HSGs) would not only imply $\mathsf{RP = P}$ but $\mathsf{BPP = P}$ as well, but I can't even conceptualize why the former holds.
The paper states that
Having such a generator that runs in polynomial time enables a trivial deterministic simulation of an RP algorithm by using each of the generator’s outputs as the random pad of the given algorithm.
Every paper or lecture notes I've found on the topic of HSGs and $\mathsf{BPP}$ derandomization also mentions that $\mathsf{RP}$ derandomization is trivial using an HSG, simply by trying all elements of the hitting set as random pads.
I am confused as to why this is the case. At least one element of a hitting set must be accepted by circuits that accept over half of their inputs. However, RP circuits don't necessarily accept over half their inputs; rather, if $x \in L$, only then will the circuit accept over half the time. I assume that fact is related to why a hitting set must contain a valid $\mathsf{RP}$ random pad, but I do not understand exactly how.
So, my question is: how do the circuits in the definition of an HSG relate exactly to $\mathsf{RP}$ circuits? i.e. how do the elements of the hitting set - which are to be the random pads across all inputs - relate to the randomness of RP algorithms when such algorithms accept over half the time only when $x \in L$?