First of all, a hard push does not make a noise.
The ref has tools to use besides putting his hands on someone. He doesn't like the kid jawing at him? Pull out the yellow card. He was quick enough to do it when the coach came out. Shouldn't he have smacked the coach as well?
Only someone who is mentally weak needs to put his hands on a kid.
I didn't go to a private school. I went to a pretty rough small farm town high school. I would feel right at home in Oklahoma.
I played competitive sports from the time I was 6 years old. I started football in 4th grade and played high school ball for the winningest coach in state history. I wrestled from 6th grade on. I wrestled for amazing coaches. There were coaches who put athletes into the Olympics, including the winningest D1 coach of all time (who was a notorious hard ass). Was I ever grabbed forcefully? Yes. Did any of my coaches, who I knew and trusted, ever hit me with a "hard shove" like that? Oh, hell no. With exception of my college coach, if they had, my father would have been right there to come after them. Then he would have kicked my ass.
Finally, this wasn't his parent, or his teacher, or his coach. This was an official he didn't know and who didn't know him. It is a completely different dynamic.