True, may the best person win. However, it's still a step the US has yet to make. (Not saying that the next president has to be a woman, mind you, just that it would be nice if it happened in the not-too-distant future.)
There is no inherent contradiction, since the term "men" can also be used as a gender-neutral synonym for humanity. It has only been in the relatively recent past that the gender-neutral meaning has been deprecated. I make no claims about the original interpretation of the phrase, but when I cited it, I chose to use the gender-neutral interpretation of it. Under that interpretation, there is no contradiction.
I'm well aware of the term men being used as a synonym for humans/humanity/humankind, and that most people prefer to read it that way in the given context. However, I cannot fathom that it was meant that way. It's more of an idealizing of a not-quite-ideal ideal.
About anybody's country being perfect: No country is, and those that claim to be without skeletons in the closet most certainly have many. (Turkey, anyone?) My own homeland has its own fallacies to overcome, both in the past and present, but like any other country, we have to play the hand we've been dealt.
