Omega Race is a lot like Asteroids. The game PCB is divided into two parts: the program side that handles inputs, scoring, and gameplay, and the vector generator which takes data the program has generated to figure out where and what to draw. The game will not work unless both of these are working. The third half of the board is the analog section that converts the results from the vector generator into actual voltages that are sent to the monitor to dictate the X and Y coordinates of the objects on screen as well as their brightness (Z). If this part doesn't work the game will 'play blind' which means coining up, sounds, and controls will be working, but no picture (and the LED on the monitor's deflection board will be lit, deactivating any drawing). Or it could be a faulty monitor just not showing the picture. But you're not there yet.
The LED on the board staying lit probably means the switcher is not actually hooked in to the power switch on the game, so it stays on (and therefore so does the PCB) whenever the game is plugged in. Wiring the supply for the switcher to the line filter will put it past the main switch (but not the safety interlocks), or you could wire it to the 120VAC supply taps on the big transformer.
Flashing P1/P2 and clicking coin counter sounds like the board is in a reset loop. First step is to remove and clean the legs of every socketed chip, as well as the edge connector of the motherboard and the smaller daughterboard. Make sure not to bend any legs while removing or reinserting, and don't scratch the board below the sockets by prying them out with a screwdriver (use a puller or be very careful). Then put it in self-test and turn it on and see what happens.