The rebooting threw me off a little too, but mostly because I try to run how I would do this task, in my mind, and programming a hotkey script to shutdown and reboot with a new game seems much more complex/prone to cause issues than a script to quit out of current game and start a new game (and change the default bootup game). I trust your ability, not so much mine.
I just bought myself a pachislo machine, so an idea on top of my mind is there is a key switch required to be turned to change the odds on the machine. Forcing a key for swapping could be one possible discouragement from rebooting.
Also, regarding the cartridges. I both love and hate it. I love the feeling of a cartridge, even if the game isn't truly present on it, I am a little bit iffy on the loose parts and relying on various connectors playing ball. That said, after reading about PL1's suggestion of roller switches, I am reminded of a board game I had as a kid. It was called "Shrieks and Creaks".
What it had was a little speaker called the talking tombstone, and you plugged it into your tape player, and just played a tape the entire time you are playing the board game. The tombstone was mostly just a speaker, but without any cards inserted, it would just cut the audio. Anyway, Each player had a player card that they would routinely need to insert into the tombstone along with another key card. When the cards were inserted, it depressed different combinations of switches, that worked just like dip switch settings, if the right combination of switches were depressed at the same time, it would connect the audio feed and you would hear the instruction of where to go on the board. If the combination was not triggered, you would get
dead silence, and are just able to finish your turn.
Anyway, my thought is that you could avoid doing anything with embedding connectors if you simply had different patterns cut into the sheet of acrylic. Depending on the combination of switches depressed from inserting the card, it would be recognized as a different game. Combine that with a key to lock it in place, if you want.