Star Trek III is all about emotions, feeling, family and sacrifice. This is why I love it.
The hero plot is simple enough:
Kirk finds out about Spock's coffin still being intact (this happens between scenes thanks to the shuffling).
Sarek and Kirk discover Spock's "soul" is in McCoy and must bring them both back to Vulcan so they "find peace." It's not explicity stated Spock's body is necessary, but it's important to Sarek for Spock to not be left on Genesis. So he decides then to get McCoy, go to Genesis, get Spock's body, and bring them to Vulcan. Sarek assumed Kirk had the Katra, so it was Spock's body being left behind which annoyed him. Why? Amanda probably gave Sarek a hard time.
Once Kirk get to Genesis, he is told Spock is alive. So now picking up Spock is even more imperative.
On the surface, after Kruge is dead, Kirk sees Spock's body is now at the same age as when he died.
On the way (again we don't see it - it's implied), Kirk tells Sarek Spock's body is living. Sarek that that time decides to do the Fal Tor Pan to attempt to restore Spock's consciousness to his own body (instead of "the Hall of Ancient Thought" - was this in an earlier draft of just the novelization? Either way I like the idea).
None of this took me 39 years to suss out. I walked out of the theater with this understanding. So for me, the story was never confusing or muddled. It made perfect sense. Here's what I allowed my brain to gloss over because I loved the experience to much to care:
Why did Kruge need to wait for Kirk to use the transporter to beam over? The BoP transporters were working afterwards. And the Enterprise transporters seemed to have initial trouble beaming Kirk's party off the ship. A line or two of dialog would have smoothed it over instead of me making a mental excuse.
Star Trek III's "plot holes" are no more glaring than Star Trek II's. Both movies are incredible experiences.