It happens with any cartridge I try, I also did swap the RIOT chip with my Atari 2800 (Video Arcade II) and I get the same results, and the old RIOT chip worked fine on the 2800, so I don't think its a bad chip. I did this with all the socket chips with the same results. Perhaps its a bad ceramic disc capacitor? Everything visually looks fine.
Those go bad pretty rarely, but anything's possible. It's also possible (and more likely in my experience) that you might have a bad socket - socket wipes can lose spring tension, can become dirty or corroded and in extreme cases completely break. I've run into that more often than bad ceramic caps (which I've never personally experienced). Given that you're having issues with a 6-switch, the most likely culprit to me is probably the ribbon cable between the two boards, or one of the connections on either end.
Anyway, if swapping chips didn't address the issue, what I would do in your circumstances is to open up the system and make some measurements with a logic probe (if available) or at least a multimeter as you try to cycle the switch, with a copy of the schematic handy while you do it. You'll figure out the issue pretty quickly.