As a feminist sympathizer, If I want women to feel empowered, shouldn't I wait for them to make the first move?

You don't have to, asking someone out is not oppressing them.

You have to be a sympathizer for a good cause, and a gentleman will. Meaning, when you see a woman is interested in you. You will be able to tell. She will smile, and be friendly. She will be herself. Not flippant. This will basically, show you the difference in one that earns your respect, or feels entitled to it. A friendly woman is more at peace with herself. The other will flip her hair around, move her leg back and forth constantly, anything to get your attention more than needed, because that is just what she is. Or, will be extremely shy, making you feel too concerned about her. She isn't that shy to be at the same place you are, it's the opposite of the other. But, basically the same ploy that will get your attention more than the other's. Better off to avoid this type. They need more of everything you have, then they are willing to give you in return. One who won't, will be one that won't make you want to feel hard pressed to go over there quickly. She's interested, but not overly so. You share a common ground. She will give you respect, and expect it. So, you introduce yourself to her, in a friendly manner, and have a conversation. She won't keep you guessing or chasing her. She isn't playing around. Because, she doesn't want to be played either.

Not if you want to date them. If a woman really likes you, knows you like her and you take too long to make the first move yourself, she will ask you out (unless she has a super traditional mindset). If she doesn't really REALLY like you or can't tell you are interested she will move on and you will feel like an idiot.

Just because woman are not clamoring to get you on a date doesn't mean they are stuck in the 19th century; it might mean no woman likes you that much (at the moment).

No. Whoever feels like it should make the first move. There are no rules.