The Gi is controlled by a Triac mounted to the power supply chassis. The triac is controlled by a flasher board on the playfiled. This sort of problem is typically caused by a combination of the following:

A problem with the load lamp mounted near the flasher board. Look for a light bulb mounted near the board that doesn't appear to do anything. Make sure it is not missing or has been replaced by an LED
Loose wire or pin connection on the lamp flasher board
Loose connection to the triac,

My Spectrum had this issue. You can swap the two Triac boards as they are the same to see if the problems moves with the board. Mine was fixed by replacing the MOC3011 chip on the board. I think the chip was cheap. Also, check the connectors and reflow the pins. These boards are crazy old!

The schematics say the two aux driver boards under the playfield each drive its own triac mounted on the power module in the bottom of the cabinet that switch the playfield G.I. on and off. Each triac will probably switch half of the playfield G.I. lamps considering the load - there are usually two G.I. lamp buses from the rectifier board to the playfield in these older Ballys to reduce strain on connectors and wiring.

One of the backbox aux driver boards does the same for the backbox G.I. (triac also mounted on the power module bottom of the game).

The fourth aux driver board in the backbox (note it is different to the other three and has a SCR mounted on a heatsink) controls four 12 volt lamps (#912 / #921 lamps) behind the backglass - see here where Todd (TNT) mentions them:

Todd mentions the four 12V flashing lights here when you get the 10k star bonus: