Why does Candy say he should have shot the dog instead of allowing Carlson to do it? Chapter 3, Of Mice and Men

In Chapter 3 Candy allows Carlson to put his dog out of its misery and shoot it. Carlson takes the time to explain where the dog should be shot to ensure the quickest death and least amount of suffering. Later in the chapter Candy states that he should have done this, that "you shoot your own dog." The shooting of Candy's dog in Chapter 3 foreshadow's events to come later in the book.

Just as the dog is Candy's responsibility, Lennie is George's responsibility. As Lennie becomes more violent, George understands that Lennie will need to be killed. George understands that this is something that he needs to do to own up to this responsibility as well as to ensure that Lennie is taken care of in the least painful manner possible.