Sorry, but I don't think a bigger power supply will fix your problem.
In the multi-switcher I am building, I have the exact problem you have, but with area 51 site 4. This game draws about 10AMPs on the 5v line. Post is here:
http://forum.arcadecontrols.com/index.php?topic=110970.0The problem is the voltage drop over the run of your "cheap" wires is too much to run the PCB. More AMPs will NOT solve a voltage drop problem. Only bigger wires or higher voltage can make up for the voltage drop.
WARNING: I am assuming you have an adjustable power supply that you can turn the voltage up on.
Voltage drop will increase as the draw of AMPs increases. So, a PCB needing 5v and 5AMPs might drop .1v over the wire, but another PCB needing 5v and 10AMPs will drop .3v over the same wire and not work due to only being 4.7v. (These numbers are made up to give you an idea of what happens.)
So, to solve your problem, I would recommend you FIRST check that ALL PCBs work when plugged directly into the 6in1 board. Anyone that doesn't work, you will have to turn up the voltage on your power supply until it does work. Remember the max number for the "worse/highest" board. If this number is too high, that could be a balancing problem of all of the boards you want to use. The trick is that you can't turn it up too much, such that the PCB with the lowest power demands gets too much voltage.
For example, lets say the most hungry PCB requires you to turn up your power supply to 5.3v to work. If you check at the board, it likely is only actually getting 4.9v to 5.0v of actual power. If you turn on the least hungry PCB with the same settings, it might actually get jolted with the full 5.3v!!! Yikes!!! (Note: adjusting the power supply inbetween switching games for "normal" use kind of defeats the point of all of this trouble putting in a 6in1.) BE CAREFUL WHEN TESTING THIS TO TURN DOWN IN BETWEEN SWITCHING BOARDS!!! Or you risk frying something.
The solution I took to this problem is to ensure the voltage drop across my wires and relays is so low, that I won't have to turn up my power supply at all. (or at least not past 5.1v)
Since you have a canned solution, you can't control all of the wire and likely none of the relays. If you have to turn up the voltage to work your most hungry game to about say 5.3v, you will have to balance out your most hungry games with more wire and lighten up the wires on the least hungry games to ensure the voltage AT THE BOARDS is with in specs FOR ALL BOARDS with the same power supply voltage. The key is VOLTAGE AT THE PCB. At the power supply is best case for voltage you will get on any PCB.
Does this make sense?
Basically, all of the PCBs AT the PCBs have to have the correct voltage with the same power supply settings. You get to vary the wire to make it work.
If plugging direct to the 6in1 works for the PCB, I would tell you to either buy a better extender cable or desolder the power wires and solder in more/bigger wires. I am guessing it is 1-18awg wire for each power line. ALSO, your ground MUST be the same size as the biggest of your other lines (12v, 5v, -5v). Voltage drop is also higher (worse) for lower voltage. You will likely only need to increase the wires for ground and 5v. In my case, I increased 1-18AWG to 4-16AWG for 5v and ground (you can see in the pictures in my thread). This is WAY over doing it, but I wanted the problem dead and done.

Let me know if you have any questions.