The thing about "why going with Edelgard" can be answered with:
1) She is apparently the most popular of the three lords, according to in game data.
She was also the most marketed character for the game, having a figma of her already in the works before the game was even out, was the bigger focus of the 3H banner in Heroes and even the main theme, Reach for My Hand, is sang from her point of view and its Japanese title is Lady of Hresvelg.
2) She is the most relevant of the three to the plot, being either the protagonist or the antagonist, unlike the other two who became secondary characters on other route beside their own.
3) That argument could be also used for
over Decidueye and Primarina or
over Chesnaught and Delphox
Many try to find a "pattern" in Sakurai's choices. People were speculating Chrom because "Look at Marth, Roy, Ike...He is going to only add the main lord", anand they're doing the same thing again. "Look at Robin, Corrin...he is going for the playable avatar".
But in reality Sakurai's methods aren't set in stone. He has stated before that characters must have a clear personality of their own for being chosen, and Byleth lacks it. Plus he/the Smash team might feel that another blue haired middle weight sword fighter avatar, even if as unique as Byleth is, might be too much and instead go for a tanky small axe user instead, especially for how prominent Edelgard is.
I'm not saying that Byleth has no chance, but that Edelgard has a way bigger chance than what people think.