Traits have consequences. Not a purpose or function.
If we take a look at our primate relatives, we see a wide variety of facial hair…mustaches, beards, sideburns, cheek patches, eyebrows…of all different lengths and all different colors. Same with funny shaped and colored facial features…noses, cheeks, etc, the proboscis monkey and the mandrill are famous examples.
And these features are almost certainly due to sexual selection. So it seems pretty likely that the distribution and quality of human facial hair has a similar cause, signalling who is male and who is female and what species they belong to. And human lack of body fur could have a similar origin.
However, I can’t help but feel that human savanna living, head hair, bipedalism, body nakedness, and copious sweatiness are all intertwined traits. We lost our body hair for some reason, but retained head hair as a sun screen. And this probably happened after we became fully bipedal, because only the head is fully exposed to the savanna sun. But the exact conformation of our beards and mustaches and eyebrows are probably determined by sexual selection, not adaptation to the physical environment.
I realized this after I started shaving my head 
As I’ve mentioned previously in these sorts of threads, our loss of body hair (which, incidently, we haven’t lost; it’s just much shorter and sparser than in our relatives) probably has more to do with our neotenous nature than any adaptive significance to losing hair itself. It’s a by-product of a trait that is advantageous. And an example of why it’s not always the best choice to go straight for the adaptive explanation.
Relative hair loss in humans is essentially a rider for a more adaptive trait. I don’t see why what hair we do retain in significant amounts can’t be similar. I don’t buy “hair as head protection”. Since we most likely evolved with dark skin, the sunburn aspect would probably already have been minimized. And if it weren’t, we likely would have simply begun covering our heads the same way we started covering our bodies when hair loss became an issue in colder temperatures.
Nonsense! Do your opposable thumbs allow you to do something that you wouldn’t be able to without them? What about your particular skeletal structure? Your pancreas? Brain? Purpose is as simple as what a thing or a trait does. It need not be more profound, and you don’t need to worry about talking about purpose or function in this sense as somehow introducing direction to evolution. Traits are consequences of evolution, which serve a certain purpose or function. Think about it. That you can walk is a consequence of the evolution of a complex array of features, each of which have some function in having you walk.
I understand and appreciate the philosophical goals in mind when you’re trying to exclude purpose and function from the discussion of the evolution of traits, but this kind of statement is taking it to a ridiculous extreme. You’re essentially depriving the discussion of evolution of one of the most important features of natural selection: its non-randomness, which exists in spite of the process’ underlying blind, algorithmic mindlessness.
To clarify, a point in my first paragraph above, it is traits that have a purpose or function (this is a generalization, not all of them do - but the adaptive ones do). If it looks like I’m saying that evolution has a purpose or function, that’s not my intention. I mean to emphasize the exact opposite. Evolution has no purpose and serves no function, but nonetheless gives rise to objects with these qualities. That’s part of what makes it so amazing.
Didn’t we most likely have light skin (like chimps) before we lost our hair? From what I’ve read, dark skin probably came after we lost our hair (or, as you correctly poin out, most of the hair on our bodies became very short).
I couldn’t tell ya whether we had light or dark skin, to be honest. And I really don’t think anyone else can, either. But either way, it doesn’t really explain the whole “full head of hair” thing, unless it has been confirmed that hair reduction occured before our ancestors learned how to use tools. A prospect I find dubious, to say the least. Like I said, if they had dark skin and no significant scalp hair, the head protection angle is pretty much a non-starter. If we had light skin which may or may not have been prone to burning, and no hair, they’d have put something on their heads, in the same way we covered our bodies to protect against other extremes of climate.
And, of course, if we had head hair from the start, then it couldn’t have evolved as an adaptation in response to potential head sunburns.