Ideal solution imo-
Women keep their name on marriage.
Men keep their name on marriage.
Children have both names, double barrelled.
That way it is easily identifiable who a child belongs with. Parents don't need to belong to each other and be identifiable as a couple to schools/doctors etc because they're adults and can speak for themselves.
I parents split, and remarry, everybody still keeps their own name. If more children come along, they double barrel as above. That way everybody shares a name with their blood relatives, including and half siblings.
When the next generation come along and couples marry, both having double-barrel names, then they choose one name each to hand down to their children. Nobody ever has to change their last name, with all the palaver that goes with it, and parents always share a name with their children.
There, all sorted.