Upon more consideration, I will adjust my earlier post with this:
To hook up a joystick to a NES-style controller for the NES Classic:
1) No need to split the ground wire (Atari joystick pin #8) into 6 wires like I originally stated. Grounding the joystick once on the controller circuit board should do.
2) The aftermarket controller I used (not the Wii Classic clone) has a specific ground side for each button (see pics). The ground trace runs around the entire perimeter of the board so that all the Dpad buttons on the left have the ground on the left, the A & B buttons on the right have the ground on the right, and the grounds for Select & Start are in the middle of them.
3) Wire the 4 Atari joystick directional wires to the 4 NES controller directional spots on the side opposite where each ground spot is.
Pin #1 - Up, Pin #2 - Down, Pin #3 - Left, Pin #4 - Right
4) Split the Atari joystick fire button into two and wire them to the A and B buttons of the NES controller like in step 3. Since on a modded NES Classic A is used primarily to select folders and games and B is primarily used only as the Fire button in the Atari games, this setup should work fine. Pin #6 of the joystick is the Fire button
To hook up a joystick to a NES-style controller for the NES Classic through an Atari Flashback (in theory, I have not done it yet):
1) Do all the same steps, 1-4, above only instead of using the wires from your joystick pins use the corresponding wires from the male joystick port to hook up to the NES-style controller, all in the same way as with the joystick. So when you plug in your joystick to the Flashback port you are essentially still connecting your joystick to the NES-style controller, just through the Flashback.
2) Take the ground wire from one of the Flashback console buttons and wire it to the NES-style controller in any grounding spot on the circuit board. The joystick port (both ports if you're using both of them) and one of the Flashback console buttons must be grounded. Grounding just one or the other will not do, they are 2 (or 3 if both joystick ports are wired up) separate entities that each need to be grounded.
3) You can do the exact same for port #2 to allow for 2 joysticks to be used (Player 1 and Player 2). You will have the controller cords for each NES-style controller coming out of the Flashback, but I don't think anyone cares.
4) Wire the hot wire from the Flashback Select button to the NES-style controller Select button. Do likewise for the Start button. You DON'T need to do this step to the Player 2 controller.
5) I THINK this step will work, but someone correct me if I'm wrong. Since my modded NES allows you to leave a game and go back to the menu (like the Home button would do with a Wii Classic controller) when Select and Down are held simultaneously, you can wire the Flashback's Power button to BOTH the Select and Down buttons of the NES-style controller. Holding down the Flashback's Power button for two or so seconds will take you out of the game and back to the menu. Again, only do this to the Player 1 controller.
6) The Flashback’s Difficulty buttons may be able to be used to set the difficulty. I’m not sure yet.
This all looks right to me and the first part has been tested on the controller pictured. I haven’t attempted soldering yet because I suck at it, but there seems to be a lot more room on these controllers than the Wii Classic clone I originally tried, so I’m hoping I won’t mess it up.
Let me know if you have suggestions/ideas that can help me or the next guy