It's total nonsense that you even got put in this situation to begin with. It makes absolutely no sense to use "they", a plural pronoun, to refer to a singular person and makes even less sense to use "they/them" instead of "you" which is a different tense and non-gendered to begin with. The entire point of pronouns is to easily be able to refer to someone, who you may not even know, without using their name and traditionally we use pronouns referring to their sex (male/female.) Tying pronouns to identity does not make any rational sense and is annoying af, but that's where we're at right now. I totally get your frustration with this annoying student making non-sense requests. Best way to handle non-sensicle pro-noun requests is to just avoid using pronouns to refer to this student all together. Just use the person's name and name only even if it sounds dumb to talk that way. I get that "you" should be fine and you'd have no way of knowing the student was against using "you" too, but once learning that, best bet was never use pronouns to refer to that person. I totally would have been tempted to make the comment you did questioning why this person only runs girls races if "they" refused to let you acknowledge "their" actual sex through pronoun use, but in hindsight, I'm sure you're aware that comment probably ended up pushing buttons and making her feel disrepected.
From the school stand point, it's easier to replace an assistant coach or force them to comply with ridiculous pronoun usage requests than to have to deal with these social justice warriors complaining they think you're discrimatory towards this kid and that the school is complicite in this. These kind of people will make it known they're upset and make the school look really bad to the public eye which could cost the principal his/her job if it blew up like that, so even though it's bs you got not-rehired because of something stupid like this, I can see why the principal did that and took the path of least resistance to appease the parent and kid. This kind of non-sense is partly why after I left a bad coaching situation (terribe school, abusive hc) I was not eager to return right away. Administration will always bow to even the most ridiculous requests. You probably dodged a bullet by leaving this school now that you know it's like this. I hope you can find a better coaching environment at another school! Best of luck!