It does seem a little arrogant to say that because something is contrary to your intuition, it is obviously wrong.

I looked over the header of the wiki article and didn't see anything obviously wrong with the argument. The exact figures they present are obviously a little dubious since they rely on a lot of assumptions extrapolated from current data, but that isn't really the important part of the argument is it. The main point of the argument is simply that, if we assume that there is definitely a discrete beginning and discrete end to human existence, and assuming that we have no good reason to suppose that we are at a particular position within that interval, then if we have an idea about how many humans there have been, we can make some pretty straightforward probabilistic statements about how many humans there will be.

I don't see why this is a controversial conclusion. Probabilistic conclusions are not particularly strong, binding conclusions after all. I assume you wouldn't object to the conclusion of this same argument if instead of talking about humans, we were talking about something more mundane like the total number of times that lightning will strike during a discrete thunder storm.