A short list of things that consistent egalitarians believe:

For the first 1970 years of Christianity, every Christian everywhere got the issue of women's ordination wrong.
Since the husband/wife relationship parallels God and His people, God submits to us.
Somewhere around 1970, we finally figured out what Paul and Peter really meant about gender roles.
Logistics dictate that the buck must stop and someone must submit. Egalitarians exclude male headship, making them closet Matriarchs.
If you read Greek, you'd know that the Bible really means the opposite of what it says. Except Galatians 3:28.
20th Century scholars know how to translate Koine Greek better than 1st Century Christians.
Paul is a hypocrite who forbids Timothy/Titus to have female elders but tells the Romans that Junia is an elder.
One heavily disputed verse in Romans 16 repealed all of the Old Testament precedent of male leadership.
Sarah called Abraham "Lord" (I Peter 3) to signify the mutual submission in their relationship.
Paul forbids a woman even to pray in church unless her head is covered. He really wants women to be in charge.
Jesus picked 12 people to lead His Church, and Zero of them were women. We should not be like Jesus.

If it sounds like heresy, that's because it is.