Red Sail is a big op, but the Tortuga branch on the East End is a 2 boat shop that seems pretty removed from the larger parts of the Red Sail operation. It appears they now offer a dive trip departing directly from Rum Point, which may be a recent or seasonal addition - at least it wasn't on their web site last I looked a couple months ago. It doesn't say if that is once a week, daily or what but might obviously be of interest if they're staying at Rum Point.
I'm sure either OF or Tortuga would be fine, both have good reps. White Sands may be too, but no one seems to know much about them. Tortuga has long had a rep as being less suited to experienced divers than OF, but I have no personal experience with Tortuga. Tortuga does mention all their morning wall dives being instructor led, with an option to be guided or do your own thing for the other dives. That's something many divers would be unhappy with, but new divers would probably be happy with. (OF states unguided diving is allowed.)
I also saw that about a Tortuga dive trip directly from Rum Point, but I don't know anything about it. My first thought was that maybe it is for cruise ship divers - because sometimes the Rum Point Club gets packed with cruisers looking for a beach day, but if it leaves early in the morning then it probably isn't for cruisers.
Maybe there are enough divers staying on the north side to make it worthwhile, especially during the busier season, but who knows?