Nope, not missing anything.
Under normal conditions (e.g a station attacked by Thargoids), doing search and rescue missions is pointless; the station transitioning from Damaged->Repair is dependent on clearing out the Thargoid threat... so if nobody does the rescue missions, and everybody kills Thargoids, the station will be semi-repaired next Thursday tick.
Meanwhile historically, before those invasion meters were a thing, stations would just transition from Damaged->Repaired the next week, regardless of what happened.
As far as anyone is aware, there are no metrics which FD monitor with regard to these things that measure # missions completed, people rescued or anything like that (indeed; there are records of people rescuing more occupants of a station than there are population within a system), so with regard to any impact on the station/BGS and what-not, no, there's no impact doing these activities.
... which is a big missed opportunity... yet again...
wasn't running rescue missions supposed to reduce the quantities of materials needed for repairs?
I remember reading this somewhere, but i might as well be completely mistaken