In Chapter 10, a rabid dog named Tim Johnson staggers down the main street of Maycomb. When Sheriff Tate arrives, he gives his gun to Atticus and insists that he take the shot. However, Atticus is reluctant to take the rifle, but Sheriff Tate insists that Atticus take the shot. Sheriff Tate mentions that he is not a good shot and knows that Atticus is an excellent marksman. Sheriff Tate is also aware of Atticus' adolescent nickname "Ol' One-Shot" and does not want to risk shooting a stray bullet that may hit the Radley home. Sheriff Tate feels more comfortable with Atticus taking the shot, which is why he gives his rifle to him. Essentially, Sheriff Tate is not confident in his abilities as a marksman and knows that Atticus is a better shot than him. Atticus takes the rifle, then shoots and kills Tim Johnson in one shot.
On page 104 Lee writes, "Take him Mr. Finch." Mr. Tate handed the rifle to Atticus; Jem and I nearly fainted,
"Don't waste time, Heck," said Atticus. "Go on."
"Mr. Finch, this is a one-shot job."
Atticus shook his head vehemently:"Don't just stand there, Heck! He won't wait all day for you -"
"For God's sake, Mr. Finch, look where he is! Miss and you'll go straight into the Radley house! I can't shoot that well and you know it!"
"I haven't shot a gun in thirty years-"
Mr. Tate almost threw the rifle at Atticus. "I'd feel mighty comfortable if you did now," he said.
I don't think that Heck Tate necessarily does NOT want to shoot Tim Johnson. I think that Heck Tate bows to the better shooter in this case. Atticus was nicknamed Ol' One Shot back in his day because of his sharpshooting skills. Heck is the kind of person who is not afraid to do what is right and that means choosing the right person for the job, which in this case means Atticus. Besides, the danger of Heck missing means that a rabid AND injured dog is still running around and Heck just will not allow that on his watch.