Yes, I absolutely think men should ask women out first. Even though I tend to like men who are more easy-going and peaceful, and not very aggressive, and sometimes I'm drawn to the quiet or shy types, I STILL think people who identify as men should always ask for the first date.
1) Women are conditioned by society to not ask men out or face a lot of repercussions (men writing them off for being easy, or labeling them as too forward, or seeing them as low value, or too masculine)
2) Women may not have as much practice asking men out because overall they are more frequently asked out then men,
3) Women on average have less testosterone, and testosterone is linked to increased assertiveness, so women are less likely to be assertive about asking you out,
4) Women are conditioned by society or nature to desire being chased by a desirable man, and it can amplify their sexual desire if a man asks them out and they see him as desirable, as many studies show. It typically would make women more stimulated if you asked, than if she asked.
5) Women may mistake your lack of asking her out as lack of interest and then write you off,
6) Women may see you as less assertive, less powerful, and less able to manifest the life you want. By taking action you demonstrate the basic ability to create a better life.
Yes, I ask out men all the time as I tend to have a stronger confidence, but I desire men less whom I ask out because of that reason, at least until they win me over. LOL.
However - I think it can be good for her to see you being socially competent before she sees you seeing her, so she can long for your approach, and get comfortable with the idea of you. Talk to some friends within eyeshot before looking at her. Then when you approach, it's good to build some rapport through conversation before you ask her out. If you shared a conversation for a while, or even a dance for a while, and she seems friendly, ask her out, and flesh out the details later.