Lemmon talking to Boom Boom like to a dog

You must have been watching a different movie. There's no scene in this one where Jack talks down to Boom Boom. The last scene with the two of them on the football field was quite conciliatory.

Walter Matthau's best line in this movie: "Why don't you kids go play on the freeway?"

Nope, I saw nothing unkind about the final scene, either; Harry was simply inspiring a talented friend to not give up on his career.

Along with the freeway line, I also enjoyed Matthau's repeated "Shut up, Mother." She, too, was pretty damn funny...hysterical, even!

Very good film that I'd never heard of before...

To the OP -- I don't see where you got that.

He has somewhat of a "I know you know better than to beat yourself up" attitude, which I see as a compliment to Boom Boom.

Maybe at the very end, where they are playing and tossing the football around? Calling "catch" and "ha" and "OK?" Is that what you mean? That strikes me as more the banter of two guys playing.

It's good that a Turkish person is concerned about possible racism. But I don't think it's there. Don't see it.

When Boom Boom thinks he has crippled for life a good man just doing his job, he feels extremely guilty and goes to great lengths to make up for what he sees as a
unforgivable sin. His football career and life suffers.

The hurt that he has caused Boom Boom, and Boom Boom's trying to console a man who isn't paralyzed for life, makes Lemmon feel terrible. He can't take it anymore at the end. He hates his brother in law for having wrecked a good man's life.

Well, I think I caught a little hint of what the OP is talking about, when Lemmon is shouting, "C'mon, GO! GO! GO!" or something similar, and urging Boom Boom to run across the field for a pass, and Boom Boom just does, almost automatically, like a dog that automatically runs to fetch what you're throwing. At the time, I thought, "Hmm, that could be interpreted in a less-than-flattering racial way" ...but I suspect it's just meant to show that, as a football player, Boom Boom has an indelible need, a inner drive, to play the game. If it WAS an unflattering stereotype of anyone, it was probably a 'dumb jock, more instinct than intelligence' stereotype.