As a private could I go to airborne or mountain school?

There was some privates in my AMWS class...they didn’t do well but they at least were there day 1, so it’s possible. It’s going to depend on where you’re stationed as well. Alaska, Drum, and Carson are geographical bases that would benefit from the course. As for Airborne. Demand it prior to enlisting and don’t back down. It’s always a option no matter what that lying recruiter says.

I went to AIT for 42A. Any one who had a 270 or higher on their PT got to go. I was an E1 when I went to airborne school.

Mountain? Rank is immaterial. unit type plays a part. Guard unit within New England? High probability of mountain school attendance since there's a good amount that fall under 86IBCT. Active duty outside of 10th or SF? Higher probability that they will send you to Alaska or not at all.
Airborne? Alaska, Bragg, and 173d are always looking for people. Be prepared to spend a lot of time rotating between those stations if you go and then get sent to one of them.
As long as you show up to your unit, prepared to work, pass a pt test, and don't be a dirtbag, you can go to SQI/ASI producing schools.\It just might take more tries through certain units who don't find them as applicable skills.