The discussion of object ellipsis distinguishes an intended object, an overt object, and a contextually understood object. A verb can characterize the agent without an intended object, indicate an identifiable object indirectly, or name it outright.
When an object is expressed, a new question arises: why place it before the verb? Such fronting can produce different forms of restriction, correction, and attention, depending on context.
The unifying principle is that deletion, mention, and order are choices about how information is delivered to the audience, not merely rearrangements of grammar.