That Logit beat LPM when Probit is the DGP is hardly a revelation. Logit is used because its functional form closely approximates Probit.

Why not make the DGP a 2nd or 3rd order polynomial and run the race that way?

You need to read more carefully. I generated the true model as a Cauchy CDF, which has much fatter tails than probit or logit. In other words, all models are wrong -- and probit and logit do substantially better than LPM.

It is silly to use a polynomial because that won't give a sensible true model, either, because the probabilities will still be outside the unit interval!