Treating undirected graphs as a subcategory of directed graphs
You are absolutely correct; that's a perfectly valid way to view undirected graphs.

Sometimes, in undirected graphs, some things become easier and cleaner to reason about.  For instance, you don't have to worry about the difference between weakly connected vs strongly connected components in undirected graphs.  Algorithms for undirected graphs can sometimes be more efficient or simpler than if we were to apply the corresponding algorithm for directed graphs.

So: perhaps some textbooks choose to follow this treatment because it lets them introduce a problem first in the (easier) context of undirected graphs, and then generalize to the (harder) case of directed graphs.  That's just speculation.