The entire debate here is whether or not Oswald acted alone. If he did not act alone, then it opens up a universe of potential conspiracy scenarios.
Oswald being a "patsy" AND it being a conspiracy are not at all mutually exclusive. Indeed, Oswald claiming he was a patsy AND the fact he was killed and silenced immediately only provides more circumstantial evidence that there was some range of assistance and coordination.
As to the shooting position. Houston Street went directly towards his perch regardless of which side he chose to shoot from. If the goal/mission was to kill the President, you will choose the highest probability shot. Shooting at JFK on Houston St. was by FAR the highest probability shot. Shooting at him while he was turning left, and speeding up to get on an expressway is NOT even close.
If Oswald was a patsy, that then confirms a conspiracy in fact existed.
Personally, I have gone from being a believer that Oswald was in fact a patsy, to being 95% convinced he did in fact act alone.
I have never believed LBJ was responsible.
The most likely group, in my mind, was CIA agents or contract agents acting outside of the agency and doing so for revenge over the Bay of Pigs fiasco.
The cover up, orchestrated by the CIA, was done because they knew that if the guys who did it were in fact caught, the CIA would be assumed to be guilty. I doubt it if anyone in the CIA shed any tears over the assassination, but they may not have actively participated, as an organization (as opposed to individuals within the agency).