I think Heck Tate is correct in his assessment that Arthur is a quiet man who would be seriously affected if his heroism came to light. Tate realizes that people all over town would want to visit him and the women would all bring him food and want to visit. It's obvious from the way Arthur acts when bringing Jem home that he wants little attention for himself. He even stands behind the door of Jem's room, almost hiding from Scout. Scout must lead him home and Arthur really says nothing to her or to anyone else. The sudden notoriety he would attain during the inquest could scare him and do emotional damage to a man who has lived most of his life inside of one house. I think Tate is correct. The best thing to do is to say Bob Ewell fell on his own knife and let Arthur Radley alone.Certainly the children have learned their lesson and it's unlikely he will ever be bothered by them again. He is now free to be their friend and that's what he seemed to want all along.
Boo is a silent man and as soon as the others found out that he saved the children, they would swarm him. However, for a long time he has been a recluse and if others would swarm him he would feel uncomfortable which is exactly why his saving the children stayed hidden in order to respect him. I think Tate did the right thing because Boo was still able to meet Scout and he did not feel uncomfortable and on top of that they were able to respect his feelings which was important.