I think there were two conspiracies - one to kill him and one to cover it up which the second didn’t understand who the first ones were and was scared shitless about starting a World War if the fingers pointed to Russia or Cuba.
I don’t believe Oswald was a patsy. He played his role. But he was a dumb, disillusioned kid who didn’t know who was pulling his strings. He had to be silenced before he could say who his comrades were.