I originally did the 2 ICs, but then had to do it again not too long after (can't remember if months or years). So I ended up just reflowing every single joint and that kept it working well over a decade. I added flux to every joint and added a little new solder, but I didn't remove any old solder. If you plan to desolder first, then I'd recommend finding vintage 60/40 solder (with lead). If all you can get is RoHS solder, then it'd be a good idea to keep the old solder as it still has lead. Leadless RoHS solder are famous for solder joints going bad over time.
Checking the caps is a good idea. I didn't do that at the time, but I've had to change out enough electrolytic caps to know they have a finite life.