Workings of a microwave oven magnetron transformer
It's quite possible there is an RC snubber or MOV across the switching device which passes enough current that your meter shows 120VAC. I can't quite make it out from your haywire setup, but it looks to be a relay with spade terminals for the power. It could also be a triac, likely mounted on a heat sink. 

In such a case, you won't see the switch functioning unless there is a credible load in place of the transformer. Something like a light bulb would work. 

Be extremely careful, even with the lethal high voltage, and potentially dangerous magnetron excised, you've got potentially lethal exposed mains voltage.