Well, they could show it, maybe without spock going down, with a story of how Pike doesn't like the Caste system, but according to Federation Rules, Prime Directive, they can't interfere in individual planets governments, even if there a member of the federation. Maybe at the end Pike asks the Federation Counsel to change its admittance rules to abolish any caste/slavery system. Or Pike was sent there to seek admittance, and Pike says no because of the Caste system, but the Counsel overrides his objection because of the need of the Zenite.
Then Kirk goes in and bends the prime directive and tries to help the slaves.