Hiya, not to discourage new answers coming in, but I think you might find the relevant part of the FAQ useful and interesting.
As an add-on question, to what point was it influenced by military practices in professional armies?
Summing up what i've heard before:
It started as an utilitarian practice that seeped into cultural fashion by young men wanting to be more like soldiers in a way.
Keeping hair short/shaven would make helmets fit better, and free of lice and other parasites, not cover your eyes and not give the enemy a hold to grab your head, and so forth, so it was not fashion per se, but utilitarianism.
Is this correct in any way, or are there far more complex reasons (such as age thinning hair, so attempting to appear older and wiser type of thing?)
The basic answer is: we don't know.
It's easy to point to this or that era as having fashionable male hairstyles we'd now consider "long" or cultural moments where women could crop their heads (which are what's given the most attention in the FAQ), but the fact is that we can find artwork showing adult men generally having shorter hair than women stretching back in the west and ancient near east for thousands of years. For instance:
Woman wearing a cylinder seal, playing a flute (with long hair, put up), Sumerian, ca. 2600–2500 BCE vs. Seated male with cup and palm frond (and very short hair), Sumeria, ca. 2600–2350 BCE
Group statue of a man and woman, Egypt, ca. 1850–1640 BCE
Statue of Kedamun and His Family, Egypt, ca. 1550–1295 BCE
Yuny and his wife, Renenutet, Egypt, ca. 1294–1279 BCE
Amphora with twisted handles, Attic Greece, ca. 470 BCE (if you zoom in, you can see that the young man has a short ponytail while the woman has a substantial bun)
Marble stele for a man, Attic Greece, ca. 375–350 BCE (three men with shorter hair; woman in the upper right with hair that is up and styled)
We simply don't have the textual sources that would let us know where exactly this trend comes from. Did it rise independently in multiple ancient cultures? Did it have one origin and spread from there? Did it originate in prehistory? At this time, it seems impossible to say.