He wouldn't have to. To deflect attention from the true conspirators, any good conspiracy would have a "patsy". Oswald claimed he was one, and it would have been easy to set him up to take the blame. I don't think he fired a shot that day. Once he realized what had happened, I think he understood his position, panicked and was quickly turned into the most famous un-tried "criminal" in history.
This was the height of the cold war. I would have thought that anyone who did what he did would have found it difficult if not impossible to reverse their defection.
If any stats exsist I would love to know how many other people were granted acess to Russia by the Russians and then so easily allowed to return to America without much fuss.
Oh yeah my two cents. LBJ was not responsible for the assasination but he and his admistration did hide the multiple shooter fact to protect the possible collapse of our government.