Significant aspects of this piece I tend to agree with.
A draft is a terrible idea. Such a program has little or no predictive value for whether the public will oppose foolish misadventures abroad or not on top of the ethical problems of compelling military service out of a putatively free people. Pay troops more or just do more propaganda if we need to encourage enlistment; don't demand service by force. Note I would oppose some form of compulsory national service in a non-military capacity too, the military angle is not the most offensive aspect of such a project. If the process is intended to establish a common cause or to work toward large national projects then making them compulsory is not a way to achieve this versus more active choices, is not morally acceptable for a free people to practice, nor is it likely to provide the best avenues of national advancement and prosperity to create centrally controlled methods of inspiring common feeling. This is the main reason I find the Pledge of Allegiance objectionable. If you want people to do things for others because of nationality, make the nationality worth doing things for without demanding that people do so by force. At least provide positive incentives, not just the sticks.
A more general involvement and interest in IR overall by the part of the public would be useful, but is not likely when the foe is amorphous sub-state actors like ISIS or al-Qaeda rather than "Russia" or "Nazis/Japanese". We keep publicly referring to a war "in Afghanistan" rather than a war "against Afghanistan/Taliban forces". I think that is revealing as to the nature of the conflict and the difficulty the public has in understanding it or the issues involved. It is as though we are playing a contest in a stadium but don't announce who we are opposed by, or what we're fighting for, what we win if we prevail, what is at stake as a basis for engagement or lack of engagement, and so on. This is all likely quite deliberate that no one can tell what we're doing and no one wants to talk about what we are doing or how.
The simpler explanation for why the public is broadly opposed to continued operations of these types is also in there: bin Laden's dead. We put a face up on the board as "the enemy" and that guy is gone now. That news when it occurred was immensely satisfying to many Americans (and Westerners of many other stripes) and was quickly glorified in film and documentaries by the invested parties in that operation to try to cover up the variety of failures, both strategic and moral, that led to that one momentary success many years later. And of course the wars and the massive systems of surveillance and state power ground on anyway. Indeed, it was barely even questioned that they would at the time persist on and on. Fighting every other battle in this that we threw up by trying to get him, or by taking unnecessary detours, doesn't interest the public because we didn't think the goal was to quell every spot of bad news on the planet but to get the guy who attacked us. It would help if more people had read the AUMF or we had called it a declaration of war, which it was. But nevertheless the public I suspect understood this as a more simple conflict because it does not have the time or patience to understand nation-building projects as a proper use for military forces (we, the public, barely support our foreign aid budgets as it is, given that it is typically a grossly inflated feature of the government's duties and a highly favored budget cut as a result).
The one part I would strongly disagree with is this: "But if we accept the premise that there’s no definition of winning, then there’s no definition of losing, either, and we forfeit the right to use either word. You can’t “lose” a race that has no finish line." - This is ludicrous logic. There are such things as no win scenarios for one. We could argue there was a definition of "winning"; kill certain people and come home by declaring victory. Most egregiously though, a war which involves massive destruction, chaos, and expense in blood and treasure ALWAYS has losers to it, even if nobody wins. This glib presumption is offensive on its face to the cost in lives and devastation to the foreign lands that we invaded and operated in, and to the cost of lives and injuries to our own troops and civilians who were dispatched to conduct those wars. These are losses in a strategic or geopolitical sense. The analogy is not even to a race. It is more like thinking you are running a race when the other person is playing hide and seek and not even understanding that the game and its rules are not what was thought, and that this is common place as a failure where a clear military or strategic objective is not laid out in advance and the methods of achieving it are vague and uncertain. It is hardly surprising that we should lose if we don't even know what game we are playing when we agreed to the game.
Committing troops and equipment and "credibility" to battle risks much, and should have a clear goal in mind when doing so for which effort may be focused. The effective conquest and unconditional surrender of Germany and Japan (and Italy) during WW2 is very clear. The annihilation of Hussein's forces in Kuwait and the restoration of Kuwaiti sovereignty is very clear. Even clarity of purpose is not necessarily a basis for deployment of military force, but it helps establish a point by which those forces can work toward and help achieve. The similarity of both Vietnam and Afghanistan (and of course Iraq) is that it lacks this clarity of goals in the defeat of particularized enemies, or that the clarity of goal has long since been achieved and its place more amorphous and insubstantial goals stand in to perpetuate a fight that serves little purpose and stands a limited chance of success or stands to do little to improve both the lived experiences of the people living in these countries or the safety and prosperity and security of our own.