The roots might be more properly summed up as "turning apart" rather than "dividing"--not a huge difference, but perhaps implying a lesser degree of separation. I can't say I have the same reaction to the word that you profess, though I'm sure you're not alone in your feelings (though you may be in a rather small minority). From a functional point of view, the word, as defined, is probably the most accurate for uses it's applied to. The other words you mention might very well fit into a long discussion of the topic, and help avoid monotonous repetition of "diversity", but they don't really mean exactly the same thing. I see it being used in attempts to achieve some of the uniting you long for, by encouraging people to intermix with those they are already separated from in some fashion. The message usually seems to be "immerse yourself into groupings of diverse people, rather than just sticking with your own kind." Looking at it from this angle, perhaps you'll find the word less disturbing. Unfortunately, the intermingling and uniting tend to be implied, while the diverseness is specifically mentioned, and I can see where this could lead to a perception that differences are being overemphasized. I think it helps to "read between the lines" when you hear people talking about "diversity".

Going off on a tangent, there was a political ad (local to Missouri or the Kansas City area, I believe), where a mother was lamenting her son's involvement with drugs etc. after attending an integrated school, saying "it was more diversity than he could handle." It was rather transparently equating diversity with undesirable segments of society. It raised quite a stink for being racist, and critics lambasted the ad writer for giving the word "diversity" a thinly veiled negative connotation. The ad was yanked in short order.