First, the primary organs that would be capable of this education are the media, who have a profit motive to stoke any and all fears (immigration, radiological warfare, world war, coffee being carcinogenic).
Second, I think this distrust goes beyond "the CDC said something wrong once, I don't trust them as much anymore." That could be solved with education. But we're seeing a more fundamental rise in broad-based distrust of any authority figure, and it's a much more instinctive response.
In a world where authority is more powerful and centralized than ever, some are bound to feel powerless. Someone's actions on the other side of the world have more impact on your life than they ever could (pick depending on your priors: a Wuhan wet market customer's appetite, or a Wuhan Institute of Virology employee's carelessness). With such an interconnected world, how could conspiratorial thinking not be appealing to some people?
Power inherently fosters distrust and cynicism. You can have as much pedagogy and education as you want: rational discourse just can't combat a fundamental instinctive reaction, a growing sense of claustrophobia and powerlessness that some people are bound to feel. And the response to this distrust? More control over discourse, over communication mechanisms[0], and ultimately, over citizens. Governments and corporations are more powerful than they ever were. Surveillance, and enforcement of rules, are easier and more automated than ever. How could our primitive, Dunbar-number brain wiring not react with distrust, or even hostility towards (some) authority figures?
[0]: https://www.reuters.com/world/europe/politician-says-germany...