Is there an analysis of World Champions' games that shows the statistics of their game? Is there an analysis of World Champion games, with the help of engines, that shows statistics like percentage of best moves played, blunders made, creative moves which computers can only see at high depths, and etc? There might be some bias in this analysis since solid players, at least to me, will play objectively better moves since the positions that arise are simpler.
Yes, there are such studies. You can check this recent paper and references therein for the older ones. Various ideas have indeed been tried in order to quantify the "complexity" of position (e. g., in terms of complexity of search tree or dynamics of the evaluation function as the depth grows), but I think you better read the papers if you are interested in details.