I agree with the OP. Weapon Skills should be bound to their weapons. Abilities should be interchangeable (most of them, but not all.) As I understand it, what we define as classes in FFXIV are merely just our skill levels with different weapons. There really aren't any classes, per-say. Although, you could argue that your rank with a certain weapon determines your overall strength which is the misleading part. So there's two ways you can look at it, and either way would be correct.

Anyways, animations aside, I think the best thing SE can do to make each class unique is to restrict weapon skills to their parent weapon only. If any class can use any weapon skill / ability I don't see how they'd ever be able to accomplish that. But who knows, everything we know now may change come June. Maybe they have better ideas for class diversity. Honestly, any change is welcome compared to what we have now.