Or perhaps the opposite can be proven. For every example where Black mimics White an advantage is eventually found.
This won't work in chess, specifically because of check. White can always force black to break the "mimic" pattern, to black's detriment. Chess is actually extremely well "designed" (by evolution as much as anything)...which is why the stalemate and en passant threads are so annoying .
"Look, if you understood the game, you wouldn't even ask..."
There are many games created that cannot answer this simple obstacle "what if the opponent just copies everything?" and they never go very far as a result.
Is three fold repetition also an acceptable way for a game to end? Coupled with the absence of the 50 move rule, both players could simply keep moving their piece(s) continually without making any progress.
Well, yes, but for the for purposes of solving chess via tablebases that is irrelevant, because they work backwards from mate.
When I said agreed draws cannot be evidence, I did not mean draws that can be claimed via perpetual check or by repetition ("look, it's a draw, I will check you forever...want to play it out?"). Also, a pawnlocked position is a draw, along the same lines as insufficient material, because checkmate has become impossible.