Use the wire colors on the reset relay coils from the schematic above to verify that those are the right relays.
If there's enough voltage for the score motor to turn it's probably not a low voltage issue. And unless the coil wrappers are burnt the relays are unlikely to be bad. The switch on the Start relay would be the first thing to try. If you have an alligator clip test lead you could try shorting the solder tabs of that switch together to get a good electrical connection and see if the relays start pulsing.
Otherwise checking that the switch actually closes and cleaning the switch contacts would be the first things to try. Use a dollar bill, a flextone or some 1000+ grit sandpaper to clean the contacts. Put it between the contacts, close the switch by pressing in the relay armature so the contacts clamp down on it, and pull it out. You just want to remove the built up dirt, not any of the metal from the contacts.