Interesting how the true radicals got left in the dust. Then again, I have my sincere suspicions about the "counter culture's" origins, but I'll leave that alone.
Frank Kameny was a brave and intellectual man. He was not a saint nor was he necessarily the godfather of gay rights. The point is that he was but one of many activists of his generation (b 1925). The Mattachine Society was influential mostly by inspiring the creation of other groups. However, in regard to your point, there were several major differences in these groups, their philosophies, and objectives and membership. Some of them were driven by self-styled radicals left over from the Beatniks. They tended to be the ones who claimed "gay liberation" as an objective. Gay liberation and gay rights would come to mean quite different things. Gay liberation was typical out-there radical crap, especially since it was joined at the hip with Flower Power and Free Love and a lot of the 1960's crappola. Most gay people found these gay-liberation groups to be undesirable and dated. So you saw "equal rights" groups arise. The funny thing is that the idiocy of the "special rights" accusation would have applied to gay-liberation but it doesn't apply to anything contemporary.
Of course the gay-liberation crowd like to take credit for the gay rights movement but they did more to harm it after any help they originally generated. These would be the "so what" and "whatever" folks who say outlandish things which the religious right scoops up and regurgitates relentlessly.
So yes, Virginia, there was a man a long time ago who said "We don't want to be equal, we just want to be left alone." He's dead now and his followers live in a commune in the mountains of Tennessee. Seriously, they do.