Trying to enforce a proxy (shaving as a sign for a not over-stressed platoon) instead of tacking real issues is exactly the kind of thinking that makes working in enterprise so tedious.
I'd rather spend my time with soldiers practicing tactics or doing comms training.
I think looking after your feet /boots would be more useful - there's a great scene in "when we where soldiers" where the CO points this out.
In other words, you try to inculcate a desire to stay shaved whenever possible, because then you can use whether shaving happens as a metric to get a conservative estimate of a troop's hygiene generally, and this is an important thing to know.
Not just that. As GP wrote: "They do it because they are told so to do. Therefore it is a mark of some basic discipline.". Discipline is paramount for managing an army in combat.
In the infantry, we were required to shave in the field. Sometimes this means choosing between dry shaving or wet shaving with cold water, but not shaving is not an option, even during combat operations. There were times I hadn't showered for over thirty days, but was still shaving to standard. I think calling it a luxury is a stretch.
Of course in some recent conflicts soldiers grow beards to fit in with the locals.
Back in the day the SAS in Northern Ireland grew mustachers to help their cover.
That kind of uniform and grooming standard is only for special forces in special operating enviroments. It almost never applies to regular soldiers. I say almost because regular soldiers attached to special forces units in special operating environments can sometimes get away with similar standards as their special forces counterparts.
Look at historic pictures of soldiers under continuous stress. Not shaving is the first thing you'll notice.
[1] Up until the Korean conflict, far more soldiers were killed by disease than contact with the enemy. That changed with anitbiotics and better field medicine (think MASH) but disease remains a serious threat.