An earlier draft of the script had the tribe beamed up to the ship, where they were terrified of being so enclosed. Aside from that, Kirk wasn't Picard - Kirk only mentioned the Prime Directive when he was about to ignore it. He wouldn't have walked away from these people. He might have, however, used his human illogic to come with a more desperate plan to stop the asteriod (blowing up a bunch of shuttlecraft and letting the shockwave move the rock maybe). He was good like that.
No matter what - Kirk would have saved those people. Period.
Consider: out of all the Class M planets in the galaxy, they plunked the Indians down on one that is in constant danger of asteroid impacts. They gave them an asteroid deflector to compensate for this, but since planets are, you know, spherical, the asteroid has to be approaching within an extremely limited range of angles for it to work.
That's assuming that the deflector we saw was the only such device on the planet. I would expect them to have provided a network of them on the surface, each covering a specific zone. When an asteroid comes into range, the appropriate deflector clicks on. The network is activated and is monitoring it in case it moves out of the targeting range of one deflector and into the influence of the next. At least, that's how I would have done it.
Also, maybe all the other convenient Class M planets were inhabited by Romans or Nazis. Wouldn't they want to put these pacifist Indians on a planet where they would be able to live in peace? Maybe all those Space Commies stayed away from this planet because of the asteroid danger. The Preservers, being smart-ish, provided the defense network. If Salish's jerky father had passed on the info to his kid before he died, Kirk's intervention wouldn't have been necessary.
If we completely ignore this issue (as I'm sure the episode's writer did) we still have to deal with the fact that, although the survival of every person on the planet depends on this asteroid deflector thingy, the Preservers in their infinite wisdom only told one guy how to get in and operate it. You'd think that they'd want as many people as possible to know how to get inside and push the "Save Planet" button but instead the Preservers only gave that info to the "Medicine Chief" and then encouraged/allowed the tribe to turn the operation of this machine into a mystical ritual. The implication is that simply saying "push this button right here" would be too complex for the primitive Native American mind, so they had to dress it up with all this mystical hoodoo so the Indians would get the point. Kinda racist.
This decision on the Preservers' part actually makes sense to me. If you are going to preserve a society and their way of life, the last thing you want to do is teach them all a bunch of advance technology. They entrusted the Medicine Chief with the knowledge, who then would pass it down to his son, or successor. Actually, the deflector was supposed to be automatic, so they most likely only told the guy how to reboot the system ("if the blue flame is busted, play the musical notes over on the second wall on your tom toms to activate the beam"). They probably even wrote it down for him and he lost the Instruction Book (Medicine Chief Ralph Hinkley). On the other hand, you really can’t give someone that primitive complex instructions about technology and expect them to master it. Christ, my mother was pretty sharp and she couldn't stop the VCR from flashing 12. It's not racist, it's just not fair for expect them to comprehend. It even took Spock two months to puzzle it out.
Of course, none of the events in the episode would have even happened if Kirk had behaved like a responsible adult and deflected the asteroid before going to the planet to dick around for no reason, but I digress..
Ahhh, Tahiti Syndrome…
