I’d figure a slyvain and edelgard link would go something along the lines of

C: Sylvain does his fuckboy routine and hits on Edelgard throughout the day whenever their paths cross, Edelgard of course ignores it until one last straw makes her snap at him, leading to him getting severely dressed down, criticized for his unceasingly licentious behavior, and ends it with saying she knows that he’s more than just a shallow skirt-chaser and that he should be more honest with himself.

B [after his brother is killed]: Edelgard notices he’s been going out into town a noticeably less than he used to prior to the mission where he had to assist subduing the bandit gang ran by his estranged brother and decides to look into what kind of mischief he may be getting himself into only to discover that he’s actually been spending studying. After a few remarks where he tries to play it off as “remedial catch-up to try and meet the minimum academic requirements so he can go out with more girls later” and getting his bluff called out for the books of more intermediate and advanced course materials that what their peers are currently studying, Sylvain eventually comes clean and tells a bit about his family’s circumstances, his dislike of the crest-based aristocratic system of inheritance, and that his brother’s death - even though they never got along and was hated by his brother for being crested - made him realize that this tragedy happened repeatedly multiple times across Fódlan throughout history and he wants to try and at least end that cycle of trauma with him and his house. Edelgard is impressed by his newfound raison d’etre and compliments him on his diligent drive to work towards his new goals, leading to Sylvain hitting on her again and asking for a date as an opportunity “to discuss things more intimately” Edelgard leaves in a huff and tells him to try again when he’s more responsible.

A [years later during the war]: Sylvain enters the scene in a town going over some kind of list of tasks he has to do to ready his troops before the army departs when a random girl approaches him and starts flirting with him. Sylvain begins to gently rebuff and turn down her advances by saying the army was preparing to depart soon and he had urgent work to do to help ready his troops for the campaign. The girl eventually leaves and edelgard enters the scene remarking that she believes she’s witnessing an unprecedented event that would’ve left all their mutual classmates from the academy shocked were they there to witness this scene. Sylvain laughs and agrees, saying war makes the impossible possible including making even the most notorious skirt-chaser grow up enough to know when to separate business from pleasure. Edelgard responds asking if he has regrets following her and the professor against his family. He responds that he’d probably would still be behaving like a stick-in-the-mud regardless of whether he stayed with his family or not because of the war, but with her he feels like he has hope for a better future in the crestless world she envisions. Edelgard replies she will do everything in her power to make that dream a reality and that after the war she wouldn’t mind going out for some tea when the dust settles down.