Derailleurs are really simple devices. Pull the cable and they move. Release cable tension and the derailleur's spring pulls the derailleur back the other way. So I think you may be on to something when you suspect a cable (friction) issue. When you upshift, the shifter pulls the cable and the derailleur moves. When you downshift, if there's excess cable friction, the derailleur's return spring doesn't overcome that friction and the derailleur sticks.

This type of problem is generally easier to troubleshoot on a bike with exposed cabling. Not only can you see the cable run, but you can pull on the exposed cable midways to see if you can actuate and release the derailleur by hand, isolating the problem without having to disconnect anything. Without the ability to do that, you may have to resort to disconnecting the derailleur cable to check for excess friction. Grab the cable and keep it taut by hand while shifting up and down. Does it take a lot of effort to pull the cable when the shifter releases tension? With the cable disconnected, you'll also be able to actuate the derailleur by hand to verify it's working properly.