Christianity, for example, was supported by blasphemy laws. Once those laws were undermined or removed its doctrines and practices were able to be challenged, and now we see greater numbers of non-believers in formerly Christian countries as a result.
There's certainly dogma in that example but not the sort you were positing.
As I've pointed out elsewhere, you've time limited the results for no apparent reason. That a lie can get halfway round the world before truth has a chance to get its trousers on does not obviate the power of truth and the power to speak truth.
This is alt history. Religious rule in the Western World came to an end through revolution. The French revolution in particular. Religious power was ousted out of systems of governance by force, civic codes were established and that was that. In the communist world, the other large sphere that is now irreligious, force was used even more explicitly. Nothing had anything to do with peasants having a discussion about religion over a cup of coffee after blasphemy laws were lifted. Secularism as well as religion were imposed by elites and institutions, blasphemy laws were lifted afterwards.
There's quite a lot of psychological evidence suggesting that first impressions stick stronger than corrections (which everyone knows who has ever accidentally believed something just because they heard it quickly), that only people who are predisposed to corrections actually believe them, and that in others corrections even augment belief in the originally wrong claim ("see, the mainstream media feels threatened by this revelation!")
There's also a good book by Bartels and Achen[1] suggesting that prevalence of wrong beliefs can actually increase with education (climate change denial is more prominent among highly educated Republicans, highly engaged people on both sides of the political spectrum are more likely to make wrong assumptions about the demographics of the other), because they have more capacity to rationalise false priors.
So there's actually significant reason to believe that more speech, by increasing engagement increases partisanship, which in turn leads to ample adoption of misinformation. Which I think makes a little bit more sense when one looks at the state of discourse in the last few years.
[1]https://press.princeton.edu/books/hardcover/9780691169446/de...