And so it did. And here's how it happened.
Note: This NES was dead and sold to me as-is. I think the game cartridges still worked, but it's not like you can't play Tetris on everything.
Step 1: Open a cartridge
Step 2: Modify cartridge and add flash drive
With the opening, use hot glue to keep the flash drive in place.
Step 3: Open and mod the NES
Use a Phillips head screwdriver to take the top off as well as the metal shield inside.
Using a Dremel again, cut a hole in the middle of the 72-pin connector and a small hole in the back for a cord to come out.
Step 4: Inserting the USB cable
Double-check to make sure that the cable is in the right spot and hot glue it in place.
Insert the cartridge to make sure that the flash drive connects cleanly with the USB cable.
If it's good, put the metal shield and plastic top back on and you're done!