Well, I think it may have to do with that Fates the supports mostly creates the characters themselves rather then the story. For example, Charlotte's story isn't really told well within the story, but in her support conversations. But in her defense, she's not a major characters, but lets talk about a major character in the story of fates. We can say Azura is one of, if not, most important character in Fates. Yet if you think about it, she really doesn't have much going for her. All you really know is that she knows the secret of Valla and is a kidnapped Nohrian Princess. And from what I've read from her supports, she doesn't have much growth in her character. This could really be implied to all of the Royal Siblings.
Spoilers for Path of Radiance Below
Now, with a character like Ike, he goes through a lot of things in Path of Radiance. First, the death of his Father, which basically sets the whole game into motion. While Corrin does loses his mother, would it really change the game that much if she survived. Then, Ike learns about the Laguz tribes, and how they are treated wrongly by humans. This makes Ike grow into a respect of the Laguz and always protects them if he can. Then much later into the game, after Ike becomes a lord, he learns how his mother died and the secret of Mist's medallion. Now, lets compare this to the story of Conquest. Corrin joins Nohr, does a few missions for his father, then starts to lead the attack against Hoshido. While there is more to this story, this is pretty much it when it comes to Conquest.
While I do understand Support conversations do help characters in other Fire Emblem games with a great story, they just really was the icing on the cake. While with Fates, Support conversations are the cake. That's why I think Fates has a bad story, but that's just me.