Here's the controller with the new connector:

This worked on its own, but there are a few annoying things:
1: "A" and "B" buttons, while they're marked correctly for this controller, are backwards from a standard NES controller.
2: Though the A & B buttons on the top row are turbo buttons, the C button is useless as it just fires both A and B at the same time.
My solution to these issues are as follows:

I've soldered the Red and Blue wires to the traces to switch the A & B buttons back to the familiar configuration - traces have been cut in between these bypasses. I did this transfer before they reach the turbo buttons so those are corrected as well.
I also cut the A & B traces to the upper C button and soldered on new connections for "Start" (Yellow) and "Turbo" (Brown) so I now have a NES Advantage-style Slow-motion button. And I don't have to change hand position to reach a start button anymore.
It's kinda nice to have a sega-like controller working on the NES, but it's still limited with respects to the fact that it's a cheap controller to begin with.
Let me know what you think - Thanks!