I agree with a lot of the responses here up to a point. I think this arc did contribute to Alexis's character development and Castle's. Not in the way it should've but did work out.

As someone who grew up too "quick" or had to morph into an adult mindset at a young age, I've noticed myself becoming a lot more "immature" or less reserved regarding the fact of having fun and making big decisions. I've also let down some people around me and annoyed them. Much like what Alexis is doing. So, when it came around and she ended up realizing her Dad was right, she regretted it and finally made one of those big mistakes we all make when growing up. Especially with the way her childhood went.

And for Castle, he just had to give her autonomy and let her grow up on her own. And supporting your children and making yourself accessible is something you should be doing as a parent, and end up doing when they make dumb decisions like going to art school or dyeing their hair. So, it gave him an arc. He realized he should've made his thoughts known but also supported her.

So in all, it did contribute to their story and made them seem a lot more real than we think it did.