Conspiracy theories will be more intense, in my opinion. Yes, the film did allow for the whole "back and to the left", "they're lying to us. look at the tape" culture. The back and to the left argument shows a profound lack of understanding on how the human body reacts to being shot, by the way, since a bullet doesn't push you and movement is based on your body's knee jerk reaction as muscles twitch, but that's for another topic. Anyway, while the film did allow for that, we also can extract the actual goings on from the film and when the bullets hit and all those things that actually went on and dissect it. I do believe the Warren Commission used at recording as well.
If you have Zapruder forget his camera or not bring it (and have those other people who had cameras as well that day in Dealey plaza not have them), you remove all of that. You will not have a film providing proof of what exactly went on and you will have to rely on the accounts of eye witnesses alongside the autopsy and ballistics reports. Everything will be an investigation in aftermath without a first hand visual record of the crime as it progressed. That will also affect the Warren commission report. The only first hand recordings you will have are the ones on the motorcycles of the police escort, which are audio and show when the bullets were popped off. Then you run into another problem, which is a problem with eye witnesses; Dealey plaza echoes. That's why the House Committee on Assassinations found in favor of conspiracy. They took the police recording and some scientists said there were more than 3 bullets, more than 3 bullets meant Oswald couldn't have fired them all in the amount of time, and hence a second shooter and hence conspiracy. The problem with that is that there were indeed only three bullets, but the bullets echoed in Dealey plaza when they were fired. So the smoking gun evidence was wrong, and hence those findings were wrong. With those police recordings, you have the possibility for that wrong finding to come about in 1963 rather than the late 1970s and to come about as the official findings of the Warren Commission.
I don't know how, if Oswald is assassinated in this scenario, you don't have it recorded on TV. The only reason Ruby got in his shots was because he was with the press. If Oswald's not around the press, he won't be shot. If he is around the press, than Ruby will have his chance and Oswald will be recorded. And Oswald committed the biggest crime of the century, so the press will be there.