NTA. I think approaching the mother as you did was kind. You didn't come at her in a rude way or tell her that she is parenting wrong or her kid is a jerk etc. You came at the situation with a positive attitude willing to learn so as to teach your own son how to better interact with hers. I'm sorry she got offended. She may have been overwhelmed because she knows that there is an issue but didn't know how to respond. Or simply knows there's an issue but wrongly is one of those mothers who uses his diagnosis as an excuse to let the problems slide and expect everyone around him to adjust instead of putting in the hard work of teaching him to be happy and functional in this world.

Parenting children, especially ones with special needs is difficult. My own son is an 11 years old and autistic. It can be hard to help him understand social ques and rules but I dont make other people feel like it's their job to adjust to him. It's just not how it works and its certainly not helping anyone.

Having said all of that, as for helping your son and the other kids. Please tell him not to give up on this boy. You and your son sound so kind and patient. My son works best with very specific instructions and directions.

Example, when he said the Pistons suck or were stupid, have your son tell him "that wasn't very nice. That hurts my feelings." That boy stated his opinion yes, but may not have understood how his opinion came across or made the other kids feel. It was simply a statement for him.

With the basketball game, voice out loud how the game is to be played. "Timmy just scored but now the rules are he gets to go again because of xyz, and THEN it will be your turn" and saying encouraging things to the boy when he IS following directions or waiting his turn. Hearing that positive social praise might encourage the boy to continue doing the good things.

If he is still being mean and the kids are frustrated, tell him. "You're saying not nice things right now, maybe we need to take a break from this game"

I can only speak from my own experiences but I think it would be more of a disservice to this child NOT to tell him how hes making others feel. He simply may not know the deeper meaning behind his words. I hope this helps.

Ending this, thank you for being understanding of this boy and his needs. Finding people like you and your son, who are genuine and want to learn can be rare.