> It makes me wonder how this started.
The seeds to the answer to your question lay in the rest of your post.
> I've yet to actually meet someone who is against vaccines. (Or thinks the earth is flat, for that matter.)
I have known anti-vaxxers, but your point stands: they're a relatively small but growing group, and a few years ago, they were a tiny group.
So we have a small but rapidly growing group of people, widely distributed, who believe in some insane things.
> But judging by the posts on Facebook, Twitter, Reddit, et al, it makes it seem like every other person on the street is believing this stuff.
Social media sites, by definition, want people to spend as much time on the site as possible. One way to get people to come to a site and stay is to have quality content. But...
> ... outrage culture loves having a scarecrow to pillory ...
This is slightly out of the context you posted it in, but this is the second to last step. Nothing holds people's attention more effectively than more primitive emotions, stuff that comes from our limbic system, a part of our brain that hosts fear, pleasure and anger.
In the not so distant past, the number of people an average person could communicate with was strictly limited. We were all highly partitioned, and information that went to the masses was highly curated.
This brings us to the final point: the social media sites actively seek to bring like-minded people together, and also directly and indirectly bring people together in a way that will create fights.
All of these serve to maximize site engagement, and thus profitability.
I'm skipping a lot of steps here, but I hope this will shed some light on the topic.