While I agree that working around 120V CAN KILL YOU...
And the contacts being exposed on that switch make me break out in a cold sweat...
It's not impossible for a home user to find, and fix problems with their pinballs.
The basic "I don't have power" troubleshooting proceeds step by step.
Do I have power at the wall outlet?
Does the plug that plugs into the wall look good or does it look as though it might need some attention?
Does the wire into the pinball look good, or does it need to be replaced?
Power goes into the pinball through the back of the machine, is the wire frayed right there, or does it look good?
On the inside of your coin door to the left is a service outlet. With the pinball plugged into the wall, do I have power to the service outlet?
If you do, use your meter leads. Leave one probe in one side of the service outlet, and put the other on one of the switch contacts one by one, that are exposed in your picture. BE CAREFUL. If you don't get 120V, put the probe that is in the service outlet in the other slot of the service outlet. Now touch your probe to the switch contacts one by one.
If you start reading 120V to one side, or both sides, of the switch with one probe lead in one side of the service outlet, you can now activate the switch. With the switch in one position both sides of the switch should read 120V.
(When the other probe is in the service outlet).
With the switch in the other position, one terminal of the switch shouldn't have voltage on it. You have just tested your power switch for your pinball.
Now, unplug from the wall, and use some electrical tape to completely cover those exposed terminals to your power switch... seriously, that's a killer!
Plug your machine back in, your power switch should be in the 'on' position (if we were measuring voltage, we'd find it on both switch terminals).
Go to the backbox, and find pins 6&7 on J2. Be careful. Put one probe lead on pin 6, the other on pin 7. With power to the machine and the switch on, you should have 120V reading on your meter.
This is the set of steps to find out if your pinball has 120V power.