I don't think evidence is a trump card, really in anything. There's plenty of evidence for a lot of things, both social/biological things (race realism, link between homosexuality/transgenderism and pedophilia, worse outcomes for race mixed children etc) and historic revisionism/government ops (holocaust gas chambers, 9/11 being a false flag, sandy hook being fake etc) yet most of these things, despite the evidence available, are fringe beliefs. That's because people aren't convinced by evidence, their beliefs are sort of predetermined. They dismiss ideas they see as problematic or implausible without looking at the evidence. The soviets wouldn't have any more evidence than we have today, so if today's moon landing deniers can't convince people we didn't land on the moon I doubt the soviets could (sort of begging the question here, but still). Maybe such an accusation coming from a world government would lend it more credence, maybe it would have the opposite effect, as the soviets were their adversary. I just think it's shaky reasoning to extrapolate from their inaction.
Also, the US government had it's own population under it's thumb as well (probably even more competently). I don't think the US would be too unwitting to undercut any discreet efforts by the soviets.