I am not a statistician, so I can't speak to the accuracy of ELO ratings across time, but as a historian, I would say that comparing players from different eras is ultimately a flawed (if fun) project because they are acting with the specific knowledge of their time period and playing with the goals of people of their time.
Given a dynamic, post-opening position, contemporary elite players would almost always beat past elite players. This is because they would know more endings, have a larger knowledge base of known positions and how to play them, and understand how middlegame positions work better (i.e. the benefits of advancing of a and h pawns are only recently really understood).
For example, Morphy would be brilliant, but he would be looking for Morphy-like combinations while Capablanca or Karpov would be looking to restrict those possibilities with solid positional play. Morphy would probably react poorly because the elements of positional play (as we understand it) had not even been articulated yet.
Different elite GMs and champions have different goals. Anderssen to create works of beauty, Tarrasch to articulate sound principles of play, Fischer to crush his opponents, etc. It's hard to measure this if the players' goals and motivations are totally different.
This doesn't even bring up the question of time control, which has not been standard throughout history. Like, how would Botvinnik do without having recourse to adjournments?
The idea of chessmetrics to try to determine a player's relative strength to his contemporaries is a good method to understand some aspects of player strength, but it is not as solid when comparing people across time, as Morphy and Philidor, for example, did not have access to really strong competition.
One would really wonder what would happen if Morphy or Lasker or Keres had access to chess theory today, but that is another question...