In the US..
IF the private school takes NO federal funding, they do not have to proved for disabled children other than what is required for access into the building. The school has to be handicap accessible, handicap parking and bathrooms, but that is it.
I worked at a private school. We had a child that was wheelchair bound, but the parents provided a "health aide" that was screened and okayed by the board. The child was not intellectually disabled and was able to keep up with the class work. The PE class expectations were modified.
Once you start with kids with ADHD, ODD, FASD, ASD...pick random behavioral issues, the heels dig in on the school administration.
There are a lot of reasons why, but the biggest is parents pay s**t tons of cash so their little darling isn't dealing with outbursts, meltdowns, or a teacher who may have five kids with IEPs and no clue how to handle them.
Education is a business. My state is talking about vouchers. It's a scam. My old school already said it will raise entrance fees that will negate what ever the state hands over. Other schools set up a rigid entrance exams/expectations that no kid with behavioral issues would last a month. People are already predicting public schools with become a SPED dump, because they have to take anyone.
Autism supports can run $40K+ for kids with Aspergers in my district. That includes a para pro, speech therapy, possibly OP and PT too.
Curious were you diagnosed before your mom tried to enroll you and just overshared your issues, or was it so obvious that the school said "hard pass"? My school wouldn't knowingly take in ASD kids. Parents would omitted that chunk of information, and hope their kid could somehow managed. If it got to were the kid was burning up the teacher's attention more than her teaching class, that is when they were dismissed.
I know we taught kids on the spectrum, but it was more "Don't ask, don't tell", and accommodations were made if the administration deemed it reasonable. If you were on the verge of a meltdown, you could quietly leave to your safe zone. The school would not pay for paras and SPED teachers.
You dodged a bullet. Publics can be bad, but when private schools don't want a kid it can be really mean spirited. Nothing worse than a principal basically saying, "Your kid's an animal. Take him/her home."