I've seen this system before, although they were push to make switches, they connect in parallel to apply power to a timer relay, the switch triggers the relay which applies power to the contactor for anything for a few seconds to a few hours depending on the timer type. If it wont trigger and pull the contactor in when you press the switch then my guess is the relay is wired incorrectly, or has been replaced with the wrong type. As I'm sure a lot of us know these can be triggered by voltage applied or voltage removed, now if the switches were normally open type then the trigger will be voltage applied, if normally closed they'd be voltage removed. my best guess is he's got power going to a relay that should have power removed to trigger it, hence why it powers up when he removes power. If it was a volts removed to trigger and he had the wrong switch then this could cause the problem he's having.