First, your information about the New York law is wrong. You're quoting the bogus information posted on a fake news site.
Here's the reality: New York passed a law in 2015, called the NYC Human Rights Law, which requires employers and other covered entities to use an individual's preferred name, pronoun and title (i.e., Ms./Mrs.) regardless of the individual's sex assigned at birth, anatomy, gender, medical history, appearance, or the sex indicated on the individual's identification. Refusing to do so can result in penalties of up to $250,000 for violations that are the result of willful, wanton, or malicious conduct. An employer who repeatedly misgendered a trans person would be covered by this law.
So, your school children example is bogus. School children aren't employers. Someone using an incorrect pronoun casually, or by accident, would also not be covered by this law, because that's not willful, wanton or malicious conduct.
I can understand why Ms. Cox feels as though misgendering a trans person is an act of violence. She's not talking about an accidental slip by someone who isn't aware of the trans person's preferred pronouns. She's talking people who intentionally misgender a trans person, as a sort of insult, or slur. Misgendering a trans woman perpetuates the idea that she's simply a man walking around in women's clothing, and is an indication to others that it's OK to treat her as some kind of freak. Misgendering is often the first step toward violence against a trans person.
Given the very high murder rate of trans people, and trans women, in particular, I can see why Ms. Cox might feel that misgendering is dangerous. At the very least, it's rude. How would you like it if someone insisted on calling you by the wrong gender, even after you gently corrected them? Wouldn't you feel as though that person was being hostile toward you?