Power transformer grounding

I am little bit puzzled in this
In many publications it is recommended to ground surge arresters with the transformer tank and this is what is mostly done in practice.
But we also know that the core of the transformer is connected to the transformer tank, so grounding the surge arrestors through the tank actually means grounding the surge arrestors through the transformer core, and we also know that low voltage windings are the one nearest to the core.
So my question is, how are these low voltage windings survive overvoltages due to lightning when the arrestors operate??
Can this be the reason to failure of some transformers??

Thanks in advance