Fixing woofer, should I replicate actions on matching speaker woofer too?

Repairing rubber surrounds on older woofers is pretty common.
Had I know years ago, I wouldn't have thrown out my old Tamanton's...
In any case, I think replacing both surrounds makes sense, as they will degrade with time.
As far as the interior wiring, I think your repairman should give it a quick look, for peace of mind, but if the speakers perform as they should, the electronics should be fine, in my opinion.
It's things like capacitors in amps that have a tendency to leak or go bad.
Crossovers and wiring are pretty stable. Drivers and coils will just go bad, if they ever do. So, just leave them alone.
Bob

 yes have them replace the surrounds and do all the drivers at the same time. its quite common in older drivers so not a big deal. as the material ages it breaks down over time. much less money to have them replace the surrounds then buy new drivers. also if one is gone now the other is most likely soon to go as well.

It should be relatively inexpensive, a good idea, and will ensure the other speaker lasts a long time as well.

Yes. Both. Or all. This is an easy DIY and common enough there's kits and YouTube videos. I always say, if a YouTuber can do it, I can do it. But whatever. Do both.