You've overlooked another way to "win" an argument: To persuade otherwise uninvolved third parties.

Typically, two people arguing are already thoroughly fortified in their opinions. Few people find argument for its own sake enjoyable and thus are unlikely to be lured into a debate they have no emotional stake in; as well, upon rising to the occasion to defend their side, their resistance to acknowledging their opponent's valid arguments will be stronger than ever.

Less-involved bystanders, however, can view the argument with a more impartial eye, and are much more likely to be persuaded. Of course, this is typically the justification made for the style of debating you argue against in this post--especially on the internet, where bystanders are plentiful and social dynamics are strongly subject to John Gabriel's G.I.F. Theory--but it's not at all clear that such an approach is actually effective for this purpose, any more than it is for persuading the opposing party.

As an aside, I can think of at least two other reasons to engage in debate; but neither derives value from actually winning the argument, and thus are irrelevant in this context.