You can’t prove that things like events having happened are correct.

You can't so solidly that this shouldn't even be discussed.

What should be is whether you can make a machine capable of reasoning.

There's symbolic logic, so you can maybe some day make a machine that makes correct syllogisms, detects incorrect syllogisms and such.

People who don’t understand how they work assume they’re actually intelligent and can do more than this. This is dangerous and should be dispelled quickly, or they believe any garbage it spits out, like the example from this post.

Sadly there's that archetype of "the narrow-minded not cool scientist against the cool brave inventor" which means that actively dispelling that may do harm. People who don't understand will match the situation with that archetype and it will reinforce their belief.