You would either need a custom-built changer similar to a cd changer or some sort of 10-way adapter that doesn't exist. If you plan on using the original Nintendo hardware, expect to pay a hundreds of dollars more just to find a way to switch games like you've explained. Neither of those methods would allow for an on-screen menu system, though. It would be like punching in numbers in an old school jukebox, and it would cost more than the rest of your machine put together.
If you want to use hardware, consider a Nintendo Playchoice 10 arcade machine. You can just plug in whichever games you want and scroll through them with an on-screen menu.
Otherwise, if software will do, just use an emulator. Nintendo emulation doesn't require a PC with great stats. It doesn't even require a PC with good stats.