Equal justice is a huge topic. The supreme court may or may not be better at it than smaller courts, but either way it’s probably not caused by the court itself. SCOTUS uses the same constitution as the rest of the country.
As for training and selection – I think the idea is justices should come from the general public instead of an aristocratic style thing where only a certain few are chosen for that position. As far as I know all law schools provide the same training and judge selection is the same nationwide, so it’s not really necessary to have specific training for the supreme court.