I completley agree that it can make you stronger to "not move the pieces, and visualize". But, I think it can be argued that 'moving the pieces around'(especially to the point where you've messed up the board, and you have to reset everthing, 2 or 3 times on just one move, I believe this sorta activity expands my mind, and increases my ability to visualize.
So, I'm not saying you should NEVER visualize, I'm saying 'moving pieces' CAN BE a tool that grows the ability to visualize.
Also, the arguement is, is it 'wrong(or cheating)' to 'do so' in live chess, on chess.com, and I say RIDICULOUS! No.

It's not natural to play a live game (whether OTB or online) with an analysis board therefore bypassing the need to visualize your calculations.
Calculation is such a core part of playing, people assume an analysis board will not be used. This is why most people would consider it cheating, because it's likely only 1 of the players is using it (unfair advantage).
In any case, playing chess without calculation is hardly playing chess at all.
Of course most people understand this, and are having fun being creative with semantics. Just don't confuse their fun of playing the devil's advocate with what's really a simple case of cheating