I used to think gendered bathrooms were silly, but my Sex & Gender & American Imperialism professor mentioned that getting separate bathrooms was an early feminist issue, which might explain (a small percent of) the TERF backlash, they see it as "men invading women's spaces" which they fought for. There's also something to be said for rape culture, our current cultural climate might make it actively dangerous for women, cis or trans, to be in compromised positions in spaces men are sanctioned to appear in. Not that men are dangerous, but that dangerous men exist and are shielded by rape culture.

What's your reasoning for sorting people into bathrooms based on their genitals?

I'm assuming you're thinking it means someone is "done" with transition. It doesn't work that way. A good portion of trans people don't have genital surgery, both due to access issues but also simply not wanting surgery. And, then, depending on where you live, not having surgery may prevent you from updating your gender marker so you don't have ID, either. (US citizens can change gender markers on passports/passport cards without surgery. Yes, this does mean you have people with different gender markers on different forms of ID.)

Signed, a transmasculine person who was harassed in women's bathrooms pre-social transition (never mind medical!). (ETA: I mention this because it goes to show this is ultimately about policing women's genders--I was seen as "woman-ing wrong" while living as a woman. I have also been harassed in a gender neutral bathroom, believe it or not.)

Most trans people never have any surgeries. I transitioned 16 years ago and I've never had any surgeries, nor do I have plans for any.

That being said, it is not comfortable for anyone involved for me to walk into a men's room. I don't look like I belong in a men's room.

I say this with respect, because you seem well-meaning, just lacking some understanding. Here's a counterexample: if I point-blank asked you to describe your genitals, you'd be uncomfortable, perhaps you'd even feel violated, and rightly so.

This is what bathroom laws are asking trans people to do, and it's not okay. It's an invasion of our privacy. I belong in the women's bathroom, and it has nothing to do with my genitals. My genital situation is nobody's business.

Moreover, anyone who enters a public restroom with the intent of hurting or violating anyone inside should be punished severely. That also has nothing to do with anyone's genitals. Single sex spaces, while well-meaning, do not protect anyone. The fact that a person could theoretically "claim to be trans" in order to follow someone into a restroom to attack them is immaterial; such a person could enter that restroom to attack someone even if single-sex spaces were enforced. The assault is the problem, not the sexes of the people involved.

Trans people are not predators; we just have to pee.