The Abrams chapters are action movies, so of course they're not giving a rounded depiction of Kirk, anymore than the TNG movies were a fair depiction of Picard.
It occurs to me that the "healing effect" might in fact be the Travelers, or some other unseen group of time travelers from some point in the future who have a vested interest maintaining events that lead to a specific point in history. It doesn't need to be some impersonal force of nature making these course corrections.
The bridge scene is even better when you realize the paladin is a DM-insert NPC, there to explain the overcomplicated puzzles, steer the plot, and keep the incompetent party from getting killed. Once they're back on track with what the DM has prepped, he says his farewell and disappears from the story.
Even Simon & Garfunkel are recognizably just "the enemy" for comedic narrative purposes, and you can easily imagine them having their own show where Andy and Lance are the heels.
Most time travel in Trek seems to result in retroactive bootstrap paradoxes. History is only theoretical until it is directly observed, with many possible histories converging onto the present, and the observation of a time traveler collapses the waveform for them into a version of history where they were integral to events.
Before La'an went back, things played out how they played out. After she inserted herself into the timeline, events changed, but the outcomes of relatively minor changes like the gun were already anchored in the future, and so they didn't significantly alter the final outcome. The path may have changed, but the destination remained the same.
The Detectorists (available free on freevee) is my current go-to for when I need some chill positive vibes. Everything about it just feels like a warm hug.
For movies, you can't go wrong with a low-stakes slice-of-life comedy. Are You There God? It's Me, Margaret and You Hurt My Feelings are two recent ones which I loved.
The Abrams chapters are action movies, so of course they're not giving a rounded depiction of Kirk, anymore than the TNG movies were a fair depiction of Picard.