>I claim that when you show students a visualization they will see either the visualization or the mathematical reasoning and the only way to know which it is they see is to ask them to explain the reasoning behind the visualization
Reading a mathematical picture for information is no different in some regards from other kinds of reading (say, reading a trail that the bad guys went down.)
There's the information conveyed, and the reader's ability to discern it. The picture itself though, does indeed convey absolutely specific information if you are up to the task of seeing it.
I've already proven this with my e^pi example. Google for solutions and you will not find that particular *proof* (the proof I explained after showing the picture, that references the mean value theorem.) At least you will not see it very often (I've looked,) because most people will not have *that* picture in mind when finding other solutions (there are many.) If the *reasoning* stood apart, as you claim, many people should have found that proof because its very simple.
That the mean value theorem is a highly *visually* intuitive fact, makes its connection with that particular picture absolutely compelling. I doubt anyone has ever found that proof without seeing the picture first, if only in their mind.
I think your argument breaks down completely when you attempt to debunk visual thinking in mathematics by bringing up cases of incorrect conclusions drawn from a picture. But that proves nothing, incorrect conclusions can be drawn from any sort of communication.