But how would they check any of that..? (Tbh, I'm starting to forget the mechanics of this series.)
If they managed to get control of a relative majority of the players who picked cards first and civilians went to the military for protection, it would likely be easier to figure out who chose a card. Plus, weapons drop when classes are chosen (with exceptions). So, it's easier to tell who chose a card during the second pick.
Anyone not under their control would already be targeted by the military and likely be weaker than players who chose a card during the first selection round, letting them guess based on the levels.
Other than that, I think they relied on people following the command and would be too intimidated to lie. They might have had people with classes to check if someone is lying too.