I could go into the inaccuracies of statistics and assumptions but lets just say, you are correct...

Men are more apt to attempt contact with a woman than a woman is likely to contact a man.
A man is more likely to signify acceptance of a woman than a woman is to signify acceptance of a man.
It has to do with selective breeding.
A man can spread his gene pool for his entire life but a woman has a limited number of opportunities to contribute hers.
Think millions of sperm to one egg.

As time wears on us, these reproductive imperatives change and the older a woman gets the more likely she is to be forthright in her quest for a partner. The focus changes from reproduction to compatible companionship.
Some men also understand this.

If you look at the demographics concerning your assumption you would find that your assumption does not hold true the 'higher in age' brackets you go. Women post menopause are just as likely to contact a man as a man to contact them.

Personally, at 56 I would prefer a post menopause woman. I am getting too old to raise a family. It wouldn't be fair to the children. I seek a woman that wants love and companionship more than a lay. While I'm aware that we would both like sex, it is no longer the driving force of a relationship. Procreation has already happened.

Most men are not accepting of age limitations. They will seek out any woman that will have them. James Doohan (Scotty on Star Trek) had a kid at 79. My own father had me at 49. How can a man be a father when ravaged by time? How is that fair to the child?

I'm not saying that women do not lust. I'm saying that men are more progressive at showing it than women