Could well be as Owen states - windings should be isolated from metal surround.. on the bench it has nowhere to run to ground and so will read OK - once mounted to game it has path to ground.

Another possibility is that the transformer is in fact on its way out - i come across this a few times with iron core neon transformers. Transformer will test OK without load but put load across it and current draw is much higher than it should be and transformer runs hot.

Just to add though that the fault above is normally created by transformer running open circuit. In other words power is applied and current is drawn through internal resistance of the windings etc. A power step down or up tranny will continue to convert 240V even when not under load. Eventually, it starts to run hotter - laminations start to break down and so the cycle continues till it burns out.
If you have a step down transformer in your machine that is close to the wattage required youll notice it gets warm even with the game switched off at game switch.... always switch off at the wall