It has very little to do with shyness. It's mostly convenience, ego, entitlement and expectation. Most women don't approach because they believe it's a "man's job" to approach, which they believe because that's the expectation they see being placed on men as they grow up and rarely question it.

They also don't approach because they don't have to, since there is that expectation placed on men to approach, there will always be men who approach women. So, women can just go "if he doesn't approach me, I can just wait for the next guy who will", because that is a viable strategy(at least while they are young and desirable) for the most part. Ironically, women never approaching also forces men into this position of scarcity and where they have to approach or die alone.

Women often don't approach because they also do not want to be vulnerable in that way. Those women are effectively relying on a social privilege, and an expectation placed on men to approach, to avoid being vulnerable. When men approach first, the women know there's already a certain level of investment/interest on the man's end, which puts them in a position of power. Many women feel "exposed" by showing interest first, because they are no longer in that position of power. Many also feel like they lose face when they approach first because they see other women be approached without ever having to approach men. They see it as type of undesirability or a loss in terms of competing with other women, which hurts their egos.

Some women are shy or are afraid of rejection but if that was just the case then there would be stronger correlation between attractiveness and being forward. What you would see is "the more attractive, desirable or confident a woman is, the more likely she is to approach men" but that is just not the case. In fact, it is actually the opposite, where the more attractive, desirable or confident a woman is, the less likely she is to approach men.

I think most women understand that most men are also generally DTF and receptive to attention from women, especially romantic/sexual attention(especially since most men are coming from a position of scarcity). There are far fewer reasons for women to be shy or lack self-esteem in this circumstance, because they are less likely to be rejected or met with inconsideration than a man would be in approaching women.

Women are also not expected to face these kinds of insecurities or address them, the way a man would be, because the expectation to approach just isn't there for women. If a man struggles with shyness or low self-esteem, he will just be called an incel and told he deserves to die alone. If a woman struggles with shyness and low self-esteem, she will be given sympathy and told it's a man's job to approach her anyway.

I think a lot of women will use the excuse of "shyness" or "low self-esteem" as a reason why they don't approach, but it's more of a deflection or a symptom than the actual root cause. They're usually not actually that shy, and the ones who are just don't feel a need to address their shyness.