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).
If Oswald acted alone, how could the CIA be involved?