Do you have a reference for Elo ratings not being Gaussian? A casual search shows lots of graphs and discussions saying it is.

Look at my reply to bhouston.

Elo ratings for active players are close to Gaussian, but not quite, they show a very clear asymmetry, especially for OTB old school Elo (compared to online Glicko-2).

The active players restriction is a big one and one I didn't assume I in my original statement.