A little remarked side effect of time travel is that it causes infatuation (Kirk, in "City on the Edge of Forever") and horniness (Spock, in "All Our Yesterdays"). La’An experienced both!
Edit: I forgot about Bashir and Jadzia in "Trials and Tribble-ations" but honestly they just seemed to be acting in character!
La’An fell head over heels for someone who had never heard of her. Absolutely makes sense. An entire lifetime of being treated differently, because everyone knows. Even if they don’t treat her negatively, they still know.
This Kirk was the first person since grade school that she met someone who didn’t know.
Plus it ties in with the previous episode where she and Number One reflect on their augments, family history, and years of feeling shame about who they are.