The extra connectors are usually referred to as "kick" harnesses, and unlike the JAMMA harness, there is no standard between manufacturers.

If it ONLY has a Jamma harness, then either the Jamma harness is non-standard, OR, there are no 3-4 player or extra buttons for 1-2 players.

Okay, JAMMA *only* refers to the standard JAMMA harness:
https://en.wikipedia.org/wiki/Japan_Amusement_Machine_and_Marketing_Association
It refers to ONLY the large, 56pin connector that is common to all JAMMA-compatible arcade / video games.
Wiring diagram, the pinout of the Jamma harness, can be found here:
https://s.blogcdn.com/www.engadget.com/media/2006/11/jamma_pinout_engadget_howto.jpg
OR, on page 2 of your Simpsons manual.

You will notice that there ARE no connections for player 3 or 4. There can't be - the 56-pin connector doesn't have enough wires.

If you look on page 4 of your manual, you will see pinouts for CN3 and CN4 - connections for Player 3 and Player 4 buttons respectively.
THIS is your 'kick harness'.

The 'Kick Harness' - the wires going to CN3 and CN4 on your Simpsons board - is what you need to make sure matches another JAMMA game you plug into your cabinet - IF you want to have 4 people play (and if the game you're plugging in supports 4 players, of course..)

There, hope that clears it up some.

So, whatever programmable board you get to plug in there, you won't really need to worry about players 1 and 2, since they're on the JAMMA harness. But you will very likely have to do some re-wiring for player's 3 and 4. You MAY even need to do some wiring in your jamma harness - the Simpsons only had two buttons per player. (Attack, Jump). If any of those other games used more buttons (Attack, Jump, Run?) you will have to add those buttons to your system, somehow.