Some supports do that like Rinkah supports that explore aspects of the Flame Tribe, Kaze/Saizo support deal with their past and some hidden war drama, Hayato supports almost always are him talking about traditions of the Wind Tribe and Orochi/Ryoma and Orochi/Kamui in which Orochi talks about Queen Mikoto.It's less the supports themselves and more that neither Awakening nor Fates bothered to flesh out their worlds so that they feel like believable places that you could feasibly see any of these characters living in. Because they don't have that base, they focus on character quirks. Fates adds the fact that most of the characters are retainers to the royalty, which adds some new dynamics because it aligns them to a faction and ties them directly to at least two other characters (the other retainer and the royal they protect).
Think about it, Arthur acts like a superhero and upholder of justice, but Nohr seems like a pretty dark, opressive place that's generally fighting for evil gain. Why would this guy be down for all this? Maybe he helps the oppressed in Nohr? Maybe he finds Elise to be a beacon of justice in a dark place? Who knows, it's not explored. As a result, he just seems like a guy who doesn't fit in the environment he was raised in.
Speaking of, I liked how many supports mention other characters and even foreshadow some things that are explained in other supports like Kagero paintings which are alluded in the Ryoma/Orochi support and fully expanded in the Hinata/Kagero support and the story about Sakura getting lost in a forest being talked in the Sakura/Hana but also alluded in other.
I really need to finish Birthright but Chapter 23 is making me almost cry (stupid critical hits)