Rather than censorship, a better way of combatting this sort of rumor would be a government agency, for example, one that would be put in charge of food and drugs, collecting and publishing k-anonymized incidences of disease annotated by such things as what drugs and risk factors groups of individuals were subject to. This would allow such claims to be evaluated empirically.
Cost of proving a rumor is false : $$$
Cost of disseminating that proof to say 90% of people fooled: $$
Cost of persuading those 90% that your proof is more reliable than the rumor: $$$$$
Cost of updating the rumor just enough so the proof it was false no longer quite applies: 1c
Also, in addition to dollar costs there are time costs which similarly favor the rumor maker over the rumor debunker.
A spreadsheet of "cross every variable with every other variable" also does nothing but produce ample fodder for people with zero stats training to conclude that a medicine prescribed for X, causes X.
Most people do not have 20 years education in anything. For an average human, you need several stats classes before you can reliably make good conclusions from data, and reliably understand how and why bad data is flawed. There is not enough time in an average human life for everyone to be meaningfully good at stats.
Not letting experts be experts just literally does not scale.
Neither does censorship, of course - if anything, it fuels the conspiracy.
What works is education, in general. The better the quality and versatility of education, the higher are the chances an alarm would ring in person's head when he reads some controversial statement. But quality education is an extremely hard and costly problem.
Uh... There are plenty of people who seem to think census results are rigged in one way or another. E.g. I think I've seen that the infamous Heritage Foundation (I know, I know) has some beef with The Census bureau over some under-counting or something like that.
If anything, I'd love to see more zero-trust approaches to government, where possible (voting, statistics, etc). No idea if that's going to work, but probably worth a try.