In theory Bachelier is appealing because slightly simpler: log returns are a bit more challenging to think about than returns. And it also takes indirectly into account the fact that OTM calls are less likely than OTM puts because of default risk, if you assume absorbing probability at strike 0.
Displaced diffusion (mixing both Bachelier and Black linearly) allows to gain x2 accuracy.
|Lognormal SABR (beta=1)|
|Normal SABR (beta=0) using Hagan lognormal formula|
|Normal SABR using Hagan normal formula|
If one look in log scale, the conclusion is not so obvious: beta=1 produces a better fit for a majority even for long expiries, but worse for a few (30% in my case) outliers.
What's clear however, is that one should never use the Black implied volatility Hagan formula with beta=0. This leaves a question on displaced SABR. Is displaced SABR is better suited than SABR with varying beta?