Have any members of the RF, except for W&H, gone to Eton? It doesn't seem that there is a family tradition to follow. The following from Wiki 😀but probably accurate enough
In the the era of QE2 they were educated by tutors/governesses.
Her children - Gordonstoun/Benenden
The York girls - StGeorge's/Marlborough (and Catherine also Marlborough)
Louise - St George's/St Mary's, James - Eagle House
Margaret's children - Bedales
So I can't see any evidence of 'Eton or bust'. William was boarding from age 8 so has already made a change for his children.
It would be a bit of a surprise if they end up at the local comp, but beyond that difficult to see how academics come into it, over other factors such as co-ed/single sex, boarding/day etc