I have been to the site and to the museum in the TSBD, for Oswald to shoot, as JFK was approaching him, he would have to place himself on the right side of the window, exposing himself fully. By being on the left side the wall would obscure him from view.
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.