Your problem peaked my curiosity - especially the part where the 7 relay fires as part of a combo with the 4 but not with the 7 switch. I thought that ruled out an issue with the 7 coil, but I don't think it does. I have a United bowler from the same era but I think the wiring is probably quite similar if not the same. The relevant parts of the schematic are inserted below.
The switch arrangement matches your Bally with the exception of 2 extra switches that allow the player to accomplish a 7-10 pickup as a result of a SDTM split. They are labeled 10B (next to the 7) and 7B (next to the 10). We can ignore those switches.
As you can see, there is a stand alone 7 relay that drops the 7, but there is also a combo 4-7-8 relay that also drops the 7. So, if the 7 gets dropped as a result of the 4-7-8 combo switch as you mentioned, that just means that the 4-7-8 relay is good. Now... we know that the 7 switch does not fire the 7 relay. So the next move would be to see if the 7A switches cause the 7 relay to fire. As mentioned before, these switches are located to the left and the right of the 5 pin. Moving the puck down each of the strike corridors will test them. If the strike corridors both drop all the pins including the 7, then your issue is with a wire/connection between your 7 switch and the relay. If they don't drop the 7, then it could be the coil or the aforementioned relay switch (shown in the schematic as the normally closed (NC) 7 RE switch just above the 7 coil). It could also be a wiring issue between the coil and where the 7, left 7A and right 7A switches come together. That would require some wire tracing on your part and even the correct schematic probably wouldn't help you much there. If the 7 coil and 7 RE (NC) switch were determined to be good, I'd start cleaning the connectors.