Apologies I wanted to correct my earlier statements. I spent a while longer playing around with this as the results didn’t make sense to me. I also posted a question about it on cross validated which got answered by a consultant statistician
In summary if a logistic regression model (and more generally any binomial model/distribution) doesn’t correctly account for the explanatory variables properly then this results in beta coefficient values which are biased towards 0. Thus I would argue not accounting for difference in Elo is likely to lead to inaccurate estimates. The degree of this bias and whether or not it affects us here is not yet clear to me.
I played around with my simulation some more and was able to replicate these results.