Good overview of this form, I'm guessing you cover 欲しがる/〜たがる in another video?
One thing I've kind of internalized but never bothered to run by a Japanese teacher is that since 〜たい gets conjugated like an i-adjective, I kind of think of it more like an adjective expressing that the object of the verb (or the action itself) is "desired," which is partly why you can use が to mark the direct object.