It really depends on how clean the games are too. Games that are totally clean will work on a slightly dirty console, but games that are slightly dirty/worn might only work on a system that's really clean. Sometimes the connectors on some games are worn so they won't make contact as well with the system, it's not that the games can be cleaned any better.

I've had the same thing happen with some systems of mine, that only certain games would glitch while others were fine. The problems went away after cleaning the system and the games too. If you don't clean the games before using in recently cleaned system, it could just get dirty again. I cleaned my system, and then made sure to clean any game I got before using it in my system. With a Genesis it should be fine for years before needing another cleaning, depending on how often you play it.

If you can't find a cleaning kit, try inserting and removing a clean game a dozen times or so. It might clean the system connectors enough so the games would play properly. This isn't a full solution, if the system is still dirty the problem would come back again pretty quickly.