I would check hearing and eyes, and see if he can be moved to the front of the classroom.
But.
"In the meantime, my child has come home from school saying "maybe I should just die" and "maybe you should send me away and get a new kid"."
My son is a happy go lucky kid, and we homeschool so I know the teacher and know that she is nice and willing to change things up to help him...and yet sometimes, if I am pushing too hard (or he perceives that I'm pushing even when I'm not, and frankly, I rarely do, what would be the point at this age?), he reacts JUST LIKE THAT.
DS doesn't read much yet, and I am NOT pushing him. DH was pushed, was sat down and somehow forced to read at 8 (back when the world was more sane and the education system realized that not everyone is a miracle and not everyone reads at a young age, but still, by 8 he had pushed them to their limits), and now, 30 years later, he is FINALLY getting to the point where he happily reads for pleasure, where it isn't a chore, and he's also reading a bit faster. I read very young, just did it without being taught, and it brought me absolutely NOTHING in this world, apart from glasses at an earlier age.
But sometimes he'll just not want to work, and I will try to get him to do the things he did the day before, and he perceives that as being too pushy, and he will push back, hard.
Since he's my only student, I can change tactics.
If I had 24 other children, I couldn't, if the tactics are working for the majority of the class.
So while it *could* be some big deal problem...it might also be that he just learns a little differently than the majority, but because he's in the educational institution, they can't change things *just* for him. (unless you get a big deal diagnosis)
You might look into Brain Gym. It's actually called osmething else nowadays, but googling that still gets results. It's a way of helping people integrate things that perhaps they skipped as children. A woman I worked with once had many of her clients on the floor, crawling, because they had gone straight to walking, and never learned to cross-crawl. As a result, some connections in the brain weren't made, but by learning to use both sides of the body and brain like that, they were able to make progress in their mental abilities. I saw many of her clients in another capacity, and they were amazed at the differences, just by simply learning to do things like that, that had been skipped in childhood. Since your son is a child currently, it could seriously help, and it would be worth a try.