Picking one and only one sheet of paper from the stack is indeed a major problem. My cheap Epson XP245 printer (about a year old) is getting it right about 99% of the time, which is definitely way better than I can do by hand, especially at that speed. Sometimes a second sheet gets pulled in, usually no big deal, they just come out together as well. High end laser printers are close to 100% but the slightest wrinkle in the paper can still cause a jam there. Shaking the stack before placing it in the machine of course also helps a lot getting it right.
Anyway, I agree: a cheap printer is probably the way to go. For that matter I'm thinking of a small portable printer, as long as it has a function that allows you to feed a piece of paper without printing anything on it. Use an RPi to control the thing - it has the USB port you need to send the command to the printer to have it spit out a single empty sheet, after which your mechanism may take over.
I wouldn't try to physically modify the printer. The mechanisms are highly integrated, and the printers themselves also contain a microcontroller to actually operate the mechanisms properly. It's most likely easiest to keep that as much as possible intact.