The first question any homicide investigator asks is “Who benefits?”
Good point. Aside from LBJ's obvious benefit of moving up to the presidency, there were government investigations going on at the time which very well could have led to criminal charges against him, especially since his nemesis Bobby Kennedy was Attorney General. Killing JFK and moving to the White House would surely solve that problem for Johnson. And, as I said, LBJ had, in all probability, killed before.
There was one critical piece of physical evidence found by a private detective years after the assassination: a fingerprint identified as belonging to an LBJ associate in the "sniper's nest" in the Texas School Book Depository Building!