This aspect mirrors the JFK assassination plot, in the way that Jack Ruby had to eliminate the loose end of LHO.

That is, what if the real Secret Service sniper failed to kill the patsy, or somehow failed to shoot at all? There is no way such a thing could be left to chance. They chose to take the (nonexistent) heat for failing to give the USSS snipers the green light, which is nothing compared to having to explain another team suddenly opening up on the patsy.

One difference from LHO is that in that case, they ensured his death by faking his assassination. No loose end there.