"It's amazing for him to say such a thing" isn't awkward I think, it just send the message that you maybe heard from someone or somewhere, that he said such a thing, and didn't hear him saying it yourself. "It's amazing to hear him such a thing" isn't correct, you need to add "say" here, "It's amazing to hear him SAY such a thing". Here you make it clear that you heard him saying it yourself. I hope it helps!

The suggestion is incorrect, it should be: "It's amazing to hear him SAY such a thing" Both sentences are slightly different. "It's amazing for him to say such a thing" suggests that perhaps you didn't HEAR him say what he said but you were told what he said and now you are talking about what he said, whereas "It's amazing to hear him say such a thing" suggests you did hear what he said and that you are talking about what he said. I hope this makes sense!