My mother is deeply into this stuff, and I have some friends through her who are lovely people aside from the part where they believe in this stuff. They don't make up a large proportion of my Facebook friends, but some of them are very loud about their beliefs.
Now imagine if you don't care a whole lot about this stuff one way or another, you get your shots on schedule because that's what people do, you never considered government conspiracies because the news never mentioned them, and you see the doctor when you get too sick because that's what everybody does. Then you become exposed to all this stuff, written very confidently and seeming to be backed up with solid facts, charismatic people, and high production values.
I think there's a good chance you'd start believing it!
I am worried though about the (further) rise of anti-intellectualism and anti-science. The effects of this should be pretty clear just by reading the news. This does worry me more than the concerns you've described.