1/ Egalitarianism doesn't want equality. It wants superiority.

In Genesis 3:16 when God says, "yet your desire will be for your husband" to Eve, the Hebrew word tĕshuwqah (desire) denotes "a stretching out after". Properly "a longing/desire for".

This mean she wants his role.

2/ Isn't that what sin does? It makes us long after something God does not desire us to do.

What is interesting is that the LXX translates "desire" as ἀποστροφή (apostrophe) which means "to turn away from" or "reject". It denotes the personal aspect of rejecting something.

3/ What this means is that Eve was going to reject God's order for her role in marriage and instead long to take Adam's position.

So, egalitarianism started at the Fall and seeks to supplant God's design for marriage where the husband is the leader, including God's roles for...

4/4...men to teach and lead the church. Egalitarianism is not about equality, but superiority.

Keep in mind because of sin Adam would be tempted to treat Eve like an indentured servant. That would not obey God's command to "love your wife".