For a lot of guys, it’s not easy to approach a beautiful woman. He’s basically taking his ego into his hands and giving her a chance to slight it.
On the flip slide, women will never (directly) approach men—unless maybe they are inebriated. Because with them, the anxiety and ego issues of being rejected are magnified 10X. At least.
I don’t think that’s necessarily the case for many women.
For countless generations, it was drilled into women that the man makes the first move. (I even remember being told that the first ‘I love you’ from a man shouldn’t be taken too seriously; but the second might be real...)
It takes a long time to get taboos like that out of consciousness.