I guess we have a different definition of "moral ambiguity." It's not that key characters are flawless, it's that, in the issues critical to the storyline, you can't see the difficulty of the choices and issues before them and how both sides might (or might not be) both a bit right AND a bit wrong. You don't really see the "bad guys" side. To me, many well done villains are characters for whom the author or screenwriter or whoever gets inside their heads to see why they don't see themselves as "villains." They may still be villains, but when you see "the why", you see another layer to the conflict; it's not just a formula.
The guys in the barber shop -- all are white guys, as I recall -- and not just Johnny's dad, but esp. him, are pretty much clearly implied to be against the coach because he's being kind to the black autistic kid. My point is, there is no attempt to show what Johnny's dad or the others are thinking: are they decent people struggling with a prejudice, whether it be directed at mentally handicapped or a racial minority? Do they have a legitimate basis for opposing the coach? Or are they just yahoos? The movie shows them as guys who try to bully the coach and then shun him when he won't give into an unspecified prejudice. There's no attempt to show their side or their concerns or their thinking, so we can see if they are reasonable or not. All we see is, they want to win, they aren't winning, and here they got this liberal coach who's being nice to a black autistic kid -- so they'll be either bullies or moral cowards. But the implication remains.
I mean, we also could discuss the saintliness of the coach (re the central issue of Radio, as opposed to the side issue of his treatment of his own family) or the broad caricature of the state education bureaucrat, but I recognize it's a movie and we have to give some room to artistic license. And, in fact, as I was watching it, I was struck by how much it looked, in other ways, like they were trying to AVOID making it a movie about race. (I'll admit, this seems tough to do, as you point out, in a movie about that era.) That why I was disappointed when it looked like they (the producers, etc.), out of nowhere, gave in to the easy implication, without analysis or character development, when it came to how the parents treated the coach.
But I don't want to look like I hated the movie, because, as I said in my post, I liked it and was impressed by the actors' work. I just think it would be a better movie if they showed the thinking behind the opposition. Maybe I'll watch it again; it's possible I'm being unintentionally unfair.