Different cultures have been counting in bases other than base-10 for all of human history. Of course a gentleman in the 18th century wasn't the first to use binary.... that's preposterous.

The Mayans, for example, counted in based 20 (supposedly because they counted on both their fingers and, thanks to a warm climate, exposed toes).

Leibniz freely admits that he took ideas from the I Ching: http://www.leibniz-translations.com/binary.htm

What is amazing in this reckoning is that this arithmetic by 0 and 1 is found to contain the mystery of the lines of an ancient King and philosopher named Fuxi, who is believed to have lived more than 4000 years ago, and whom the Chinese regard as the founder of their empire and their sciences.2 There are several linear figures attributed to him, all of which come back to this arithmetic, but it is sufficient to give here the Figure of the Eight Cova, as it is called, which is said to be fundamental, and to join to them the explanation which is obvious, provided that one notices, firstly, that a whole line — means unity, or 1, and secondly, that a broken line -- means zero, or 0.

The Chinese lost the meaning of the Cova or Lineations of Fuxi, perhaps more than a thousand years ago, and they have written commentaries on the subject in which they have sought I know not what far out meanings, so that their true explanation now has to come from Europeans. Here is how: It was scarcely more than two years ago that I sent to Reverend Father Bouvet,3 the celebrated French Jesuit who lives in Peking, my method of counting by 0 and 1, and nothing more was required to make him recognize that this was the key to the figures of Fuxi. Writing to me on 14 November 1701, he sent me this philosophical prince's grand figure, which goes up to 64, and leaves no further room to doubt the truth of our interpretation, such that it can be said that this Father has deciphered the enigma of Fuxi, with the help of what I had communicated to him. And as these figures are perhaps the most ancient monument of [GM VII, p227] science which exists in the world, this restitution of their meaning, after such a great interval of time, will seem all the more curious.

The article wasn't so much weak, as it was in awe of an accident of hindsight. (It only looks "special" because we settled on binary for computers.)
It explicitly made the point that base 10 was used except to refer to large groups.
Their "special words" took hold only after they ran out of fingers.

In fact, if you look at it as counting the number of "bodies worth of fingers and toes" it looks less like using binary and more like "We can't count that high, but there was one fish in the pond for every finger and toe of each person in our boat). After that they just counted boats.

Its really not much different than westerners counting in dozens, and grosses (something that wiki unconvincingly attributes to the convenience of 12 having many divisors [wikipedia.org]. From the same article you learn there were Latin terms for groups of 15, 20, etc. It seems that special, extra ordinal counting numbers for baskets full of stuff are not that unusual.