Why would whoever was behind the plot use these two nut cases?

Well, I'm not sure Oswald was a nut case. Without going into detail, I'm convinced that he had a very close association with the CIA, may even have been working for them at the time of the assassination if not prior to it, and as he himself asserted, may have merely been a "patsy". As a matter of fact, I have serious doubts as to whether or not he was even a shooter at all that day.

I don't think the mob was behind the plot itself, simply because they did not have the means to engineer the cover-up, a principal part of which was the autopsy. The reason that there was a cover-up was because our government was in some way involved. If the mob was behind it, or the Cubans for that matter, why cover it up? But just as the mob and CIA worked together on their plots to kill Castro, it doesn't take much of a leap of the imagination to believe that the CIA was behind the plot, and used mob assassins to bring off the actual shooting.

Unfortunately, we'll just never know for sure.