You don’t need that little PCB per say but you might want to use the two buttons off the board.
Your 8 arcade buttons, start, select and the joystick wires directly to the Brooks Board.
With the buttons you have your options of using the terminal strip or not.
If you want to use the terminal strip, cut the ribbon cables from that tiny board and wire the appropriate wires to the correct screw terminals on your Brook board.
You can do a similar route with your joystick, of if you think if it helps buy a Sanwa Joystick wire harness so you can color code your directionals.
If you want to use the TE board for the Home and turbo button (I would wire the turbo to the touch panel key) you have to solder one wire for home, one for the touch panel key, ground and Vcc/5 volts.