I'll try to find the Sega System E instruction manual, or ideally some schematics to see what's going on there. Another thing that comes mind that is different in the MAS (besides the use of the microcontroller) is that it probably connects GND to pins #e and #27 on the JAMMA connector, wheres the HAS uses those as the duplicated button 6 output for the Pandora's Box PCBs. Try joining those two pins with the GND on the HAS side, unless that Sega System E to JAMMA adapter already does that.
One more measurement - resistance between the button input and +5V on the Sega System E (the PCB mustn't be connected to anything). Perhaps it would require external pull-ups.
One more edit: It's the octocoupler, so most likely there will be no pull-up at all on the PCB game end, the button input will be connected directly to the octocoupler. If that is the case, please put a 1-2K resistor between button input and +5V. Of course the easiest way would be to do that on the JAMMA adapter you have, or the HAS; don't do it on the PCB game.
I couldn't find the Sega System E schematics. What is the value of the current limiting resistor used for the octocouplers? Usually that's around 220-470R.