One thing is quite clear though, they wish to remove gender and place all importance on identity on the assumed at birth sex of a person. They place genitals at the absolute helm of how one can define themselves. They place limitations on identity. No vagina; no womanhood. They wish to abolish gender in favor of placing strict restrictions, basing a person’s identity on their genitals.

If someone is born with a penis, they are male, no exceptions, even in a post-gender society. If someone has a vagina, same deal, only they are a woman. All women are assumed to have vaginas, anything else is considered wrong. Where various intersex people fall into this, has yet to be seen. They seem to be erased in this entire discussion, unsurprisingly. So, why isn’t this exactly radical feminism?

TERFs want to classify who is a woman solely on their genitals. This is something that is already done by the patriarchy. People are classified by the assumptions made on their genitals at birth. These identities are considered unchangeable and part of someone’s identity. This guess at birth is then held as all powerful over the actual identity of a person. TERF ideology that a person’s gender (or in their eyes) sex, is just what the patriarchy already does. It assumes a strict identity based on genitals that are then used to hold power over all aspects of a person’s identity.

Everything about a person’s experiences in the world are whittled down to their genitals. Their experiences as a woman are only valid if they have a vagina, from birth. Their experiences with life are only valid, regardless of how society sees them, based on their genitals. This type of ideology holds that genitals are the only defining matter of a person’s identity in the world.

Question about this, doesn't it miss the point? For example, this part:

Their experiences as a woman are only valid if they have a vagina, from birth.

Yes that's true, but isn't it because TERFs literally just define "woman" as "person with vagina"? They want to get rid of gender, and if you did that, the only "gender" definition you'd have left would relate to body parts.

Or this:

If someone is born with a penis, they are male, no exceptions, even in a post-gender society. If someone has a vagina, same deal, only they are a woman.

Yes, because that's the definition of man/woman in their mind. They don't care what you wear or how you act, they are just using the physical definitions instead of attaching gender to it. This feminism has no problem with trans people acting in whichever way they'd like, they just refuse to acknowledge gender (as constructed by society) as real at all, so they stick to a physical definition.

Am I reading their arguments wrong?

This is a theory question by the way, not about whether they are transphobic in practice. Their "woman" only spaces and stuff I don't agree with, but I'm not sure I see a problem with their stance on gender abolition (at least for a future ideal).