Below you find updated graphs representing the same data.
1) bin-size is 4 elo-points
2) minimum bin-size is 4 samples
3) in the elo-delta graph I shifted all models by 20 elo points to compensate for the white to move advantage
4) I added the cdf of the normal distribution with sd=250
5) I added the function hgm suggested to estimate draws scaling by 40/25

Agreed, the gauss function is a better fit than either the linear function or the logistic function.
Looking at the new graphs I am not so sure about hgms suggestion regarding the progression of the avg-elo score function. You are right that the next step is to take elo-delta into the equation.