I think that this is a complex issue with a lot of layers, actually. I believe that to some degree, women in our society have been fed a diet of fairy-tale sort of notions that we are the object to be pursued. Some take it to the extreme to mean that they are a PRIZE and can therefore justify treating other people - men and women alike - like ****. I call it the Princess Syndrome.
Certainly, not all women are like this. I do think that many women enjoy being pursued because it is a tangible clue that someone finds them desirable. Unfortunately, I think that there still exists a double standard in this respect. Women who make the first move are often viewed as either desperate or slutty.
The few times I ever made the first move on a guy, and it was reciprocated, it was viewed only that I was DTF and that was it. Both times, I really liked the guy as more than just some sort of easy ****, but the fact that I made the first move, I think, cast me in the role of sexual aggressor instead of a "nice girl" that they'd want to date. End of story.
Sure, lots of men say that they enjoy a woman being bold enough to make the first move, but do they truly view them as some nice girl reaching out and showing interest, or just some desperate or slutty chick who wasn't worth their time because they weren't already the object of pursuit by other men? As though there must be something wrong with them since no one else wanted them? Also, sometimes when we make the first move, we get rejected too, just like you. No one likes that, I get it.
Now, I am content to let the man make the first move, but I don't play games. If I like him, I don't cloud the whole thing in mystery and make him wonder.
And FTR, I open doors for people, men and women alike, because it is the courteous thing to do and I am a courteous human being. However, I don't EXPECT anyone to hold the door open for me. That's an entitled princess mentality. Although those chicks are out there, they are not the norm, but they make such a splash, I guess that they blind others so that you can't see the rest of us who are decent human beings.