I think the burden of prove in all cases rests with the person who is trying to obtain agreement with their proposition.
This is obviously false, but I guess it's a useful POV if you are a flat earther or something.
No. As you usefully pointed out earlier, I am a duck, and ducks assiduously avoid taking a position on the shape of the earth or any other geometric controversy as a matter of policy. They avoid it like the plague, as a matter of fact.
For that reason, the burden of proof can never rest with the "deniers" of anything, as they by definition aren't positing anything, they just aren't accepting what you have failed to sufficiently prove to them.
So then, to you, someone that doesn't have any education, doesn't understand anything, doesn't try to learn anything, and simply says "show me" to everything brought before them...that person would have to be some kind of hero. I would not buy that is almost anyone's actual position, but in your case, I could buy it. You'll seemingly take any position as long as it's counter to authority, expertise, and anything established.