Here’s an alternate take. Keeping in mind that La’an is a security officer trained in investigative methods.
La’an could hear the voices and footfalls of a security response team. She knew they would be there in moments.
The investigators would be looking for the gun that shot Kirk and 3 guards.
With that weapon recovered, the search would focus on the missing shooter who they would have on security cameras in the foyer.
Without the weapon, there would be another level of scrutiny, and such scrutiny might draw attention to the Noonien-Singh project that might be enough in itself to alter history.
By leaving the weapon in plain sight, with her own markered DNA on it as well as Sera’s likely-tweaked Romulan DNA, she’s left enough for the investigators to look beyond the facility for their escaped killer without actually revealing how and where she and Sera exited.
"And then this lady left a gun in my room and very pointedly told me I'm right where I need to be."
It was an odd shot too, like the camera focused on her putting the gun down. It felt like it was meant to, or is meant to in the future, come back, but maybe it's just a dangling thread.
What if La’an leaving the gun in the room was a predestination paradox? Leaving it there means Khan rises to power, fathers more children than he would have otherwise, thus resulting in La’an being born in the first place.
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 Romulan agent in this episode directly references a sort of self-healing aspect to the timeline, like minor (Butterfly Effect) changes don’t do the job due to … something. As such, the presence or absence of that gun is probably indifferent to Khan’s outcome.
Though one wonders if her speech to him might have reinforced the malignant narcissism that presumably underlies Khan’s later cruelties.
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.
If you think about time travel seriously, you start to see how truly complicated it would make things. If the multiverse exists and time travel also exists, then inevitably there would be meddlers from the future messing things up. And following that would be those who believe such meddling is dangerous, because it creates more timelines branching off that would accelerate the meddling, until there are more polluted timelines than “natural” ones, creating a future filled with bizarrely chaotic outcomes all branching from the recursive meddling. And so there would have to be some kind of ongoing arms race of preservationists versus people who want to stack the future with timelines friendly to their own cause.
