I don't like MPs sending their children to private schools either, but I do think that Grammar schools are a separate issue.
Grammar schools segregate children at far too young an age on the basis of an outdated I.Q. test.
Intelligence is not fixed at age 11. There used to be a 13+ that children could take (which was also far too early in my opinion) but passing that meant changing schools and losing friendship groups.
Far better for all children to start at the same secondary school and classes can be streamed to cater for different levels of academic ability, as the children develop.
Grammar schools also cater for the same level of academic ability across all their subjects, this may not suit all children, who may be gifted in say, the arts, but not the sciences, or vice versa.
In a comprehensive school, streaming would also take this into account.