So instead of growing fur we take it from other animals, or make substitutes for fur. Since some kinds of clothing (bikinis, desert robes, parkas, spacesuits) can be exchanged for other kinds we don't need fur.
Human's had already lost their thick body hair long before clothes were invented. The modern pattern of human hair didn't evolve because of clothes. The most likely explanation is the need to stay cool while running (though there are other possible explanations).
Unlike most mammals, humans are adapted for long distance running (as opposed to sprinting) and early humans (
Homo ergaster or thereabouts) are thought to have hunted using the persistence hunting method (still in use by some modern hunter-gatherers, e.g. !kung San) where you run after your prey repeatedly, not by sprinting, but by keeping on tracking it down never letting it rest for hours or even days until it dies of heat exhaustion. Humans' bipedal posture and pattern of hair growth (thick hair on the head, not much hair on the rest of the body) combined with the fact that we sweat over our whole bodies (most mammals don't) is perfect for keeping cool under the midday sun, giving humans a big advantage over the animals they were hunting, who were a lot more susceptible to heat exhaustion. Most mammals stay out of the midday sun, but it's thought early humans learned that other animals were easier pickings if they hunted at midday.
This sort of thing probably goes back nearly 2 million years and is lower palaeolithic. Humans didn't begin to process animal skins until the middle palaeolithic era and it's likely that the earliest processing of animal skins was more for carrying things (babies, gathered food etc) than clothes. The Neandertals very likely made clothes. They lived in Europe during the ice ages (sub-arctic climate) so it's not really surprising, but it wouldn't be a case of them losing body hair because they worse clothes. It's more likely that they re-evolved thicker body hair and wore clothes as well (in addition to various other cold-adapted traits they evolved) because it was bloody cold. They didn't have needles or any evidence of stitched clothing. The more warm-adapted, probably less hairy early
Homo sapiens invented stitched clothing.
Even though there are other possible explanations given for why we evolved the pattern of body hair that we have, there's no doubt about the fact that it evolved long before humans started wearing clothes. Same as how hominins were fully bipedal millions of years before they started making stone tools (contrary to the earlier belief that our ancestors became bipedal to free up the hands for tool use). Human evolution is a very fast moving field and I personally would recommend being very careful that any book or article is based on up-to-date information (by up-to-date I mean from the last ten years or so).
Regarding your point about the name of our species, I've always considered that our species is anything but wise. I think it's hilarious that we named ourselves that, especially having the subspecies name "sapiens" as well, making us wise, wise man, as though saying it once wasn't enough. In my story about the cloned Neandertal, he regularly takes the blatant piss out of
Homo sapiens for naming themselves that when it so obviously isn't true. I once wrote a blog post where I gave our subspecies the name Anthropos ingeniosa arrogans (arrogant, ingenious man) - we are ingenious. We're just not wise. And we're pretty arrogant.
It's extremely unlikely that
Homo sapiens sapiens is going to be renamed. The fact that we're anything but wise is not a reason to rename a species. Species only get renamed if it turns out they're not really a species after all, for example
Homo habilis,
Homo ergaster and
Homo rudolphensis are now thought to be a single species which is a lot more varied than previously thought (as evidenced by the amount of variation found in a single archaeological site (Dmanisi in Georgia - the Georgia in Europe that is). If they all get subsumed into a single species then obviously some will be renamed, but they won't get a new name. They'll all be reclassified under one of those names. I think the tradition is to use the name that was coined first in these circumstances (biologists are very traditional about these things) though IMO
Homo ergaster is the most fitting name.
And as Helix mentioned upthread, there are "lumpers" and "splitters" so there is debate about whether these should all be subsumed under one species - I'm in favour of lumping them.
Fitting doesn't come into renaming species. Common chimpanzees are
Pan troglodytes. Troglodytes means cave dweller, and chimps don't live in caves, but no-one cares about that. It's their name and they will only change it if they turn out not to be a species.
Pan paniscus (bonobos) got a new name when it was established that they were a different species. If you identify a species or genus, you get to name it.
Homo erectus started off being called
Pithecanthropus erectus but when later it was determined that it should be in the same genus as modern humans, it got renamed as
Homo erectus. Similarly, there's some debate about whether Neandertals should be
Homo neanderthalensis or
Homo sapiens neanderthalensis - but that's not about the name, it's about whether they're technically the same species as us (same species, different subspecies) or a different species altogether.
Anyway, due to the established rules and traditions surrounding the naming system for species, our species name isn't going to get changed.