Small resistor on the plug into TC1 -- pullup resistor of 3.3K to the Read/Write signal. Some MPU boards run fine without it, some will kind-of work without it, and some refuse to work at all without it.
So... if you leave the connector on the TC1 40-pin DIP and unplug the 7-pin connector from the reset board - does they work then?
If it does work, you need to add a 3.3K resistor to bottom of TC1 between pins 7 and 11 and then leave the plug off of TC1.

Other way around. If you leave the plug installed on the TC1 connector and pull the 7 pin plug -- the game should still boot. In fact, some games would not boot at all or ran intermittently without the DIP plug installed on TC1. This was due to an overloaded Read/Write line (another poor board design). The DIP plug had a pullup resistor installed that would add extra pullup current to the Read/Write line.

But as the last poster said -- if you leave it completely hooked up and you have problems, there is a good chance that the old electrolytic caps are dried out. As caps dry out, they decline in capacitance value - resulting in shorter and shorter watchdog periods.

Try the 3.3K pullup resistor on the boards that insist on having the reset boards installed. I have seen this resistor fix this exact problem.

That's not right...the only thing connected on Genesis is the IRQ (twice), reset and power. Disconnecting from the module at the 7-pin plug should do nothing other than reflections on a very slow O/C IRQ. If you have a fading RIOT on the edge - I can see how the IRQ antenna might have an affect...but have never seen or heard of this happening.

I've seen several boards that did not run or run right until the 3K resistor mod is done -- which is on that 40 pin DIP. You can see it towards the middle of your manual's page 23. This fixes an overloaded Read/Write line. The 3K adds quite abit of pullup and fixes lots of problematic boards. To test this theory, we install the 40 pin DIP plug -- if it fixes things then the 3K is what's needed. The original poster's comments describe the exact problem we see when it needs the 3K pullup.

This is in reference to the Read/Write pullup resistor that Gottlieb added to the 40 pin DIP adapter. Games will work without the reset board but some boards do refuse to work without the Read/Write pullup resistor.

The 3K resistor is there for two reasons:
1 -- it provides a stable Read/Write signal at board power-up. There is a tiny fraction of time at power up where data can be written into Z5 but due to gate Z36A, this data write is prevented. The resistor does provide a proper biasing to the Read/Write line at time of powerup.
2 -- more importantly - A rockwell 6502 has a minimum VOH of 2.4volts, a 4069 requires a minimum VIH of 4 volts.
In other words:
The Read/Write output from U1 (pin 34) guarantees the output will be 2.4 volts or higher for a logic 1.
The 4069 located at Z36D (pin 9 input of Z36) requires the input to be 4 volts or higher for a logic 1.
The 3K pullup resistor pulls the Z36D input closer to the required voltage range.

This is fixed with a 3K resistor between pins 7 and 11 on the bottom of TC1 on the CPU board and you can see the photo of this mod on the link provided by ForceFlow.
Do realize that even though this problem exists on all System 80/80A/80B boards but only a very small fraction of these Gottlieb boards exhibit any symptoms. The chances that you have one if these that exhibit symptoms is quite slim.

LOTS of things can cause the board to not boot. Starting with the easiest:
Do you have good 5V power to the board with solid power and ground connections?
Is there any battery corrosion on the board?
Is the shared Interrupt Request line (IRQ) being held low by a bad RIOT?