I'm curious, is there any legible video on the screen when you connect a faulty board or does it just turn blue?
I have a suspicion that the processor, the 68701, isn't running or coming out of reset so after checking voltages I'd test voltages to the CPU, pin 21 should have 5V.
Pin 6 is the reset input, it's active low so if the pin has a low signal on it then it's in reset, if it's pulled up to VCC then the chip is free to run.
Pin 2 and 3 are the clock crystal connections, it's possible that probing one pin with your multimeter would give you an indication if the clock is running, if it is I'd expect to see some indeterminate voltage on there, likely to be half VCC or lower but not zero, probing one pin will almost definitely make the clock stop, probing the other probably won't but you'd need to compare with another working board.
Be VERY careful not to short anything out.