As well, they got a GUARANTEE from Kennedy that we would cease efforts to topple the Communist Castro government in Cuba. Pretty good chess move on Nikita's part, I would say.
Also, I seriously doubt that the KGB would have used an assassin with such clear and obvious ties to the USSR for an act that could easily end with their cities in smoking ruins- IF the American people really thought that they had murdered our President.
I think that, of all the possibilities, a Soviet-backed political assassination is the least probable.
It is not impossible that Oswald was simply a nutjob who managed to pull off the crime of the century- but I still like LBJ as the prime mover in this. Probably because I hated the SOB.