Most of the things that most conspiracy advocates cite to dispute the official story of the JFK assassination have nothing to do with Oswald.
The preposterouness of the Single Bullet Theory is, by far, the most frequently cited reason to doubt the official story, but it has nothing to do with Oswald. It only concerns the idea of a "lone gunman". That's all it disputes. It's the calling card of Dr. Cyril Wecht, and he has never defended Oswald. He rarely mentions Oswald's name. You can't get to Oswald innocence by challenging the Single Bullet Theory because, theoretically, Oswald could have been one of several shooters.
What else do people commonly cite? They cite witnesses like Gordon Arnold and Ed Hoffman who had encounters with assassination operatives, but again, what does that do for Oswald? Nothing! Again, he, theoretically, could have been one of several shooters.
So, what else is there? There's the issue of the rifle because Buell Frazier described Oswald's bag being only two feet long, which is not long enough to contain the disassembled Carcano rifle. But, Buell Frazier hasn't been the most consistent and reliable witness. He's changed his story many times, including his own location during the shooting. The other side is saying that Frazier just isn't remembering it right, that the bag was a bit longer. And then, they pull out the Klein Hardware invoice, etc., and the result is that there is no getting Oswald off the hook that way.
So, what else is there to specifically exonerate Oswald? And whatever else you think you have, if you follow it up with, "We don't know where he was at the time of the shots. He wasn't on the 6th floor; he wasn't in the doorway; we just don't know where he was." how is that going to go over? How's it going to sell?
Well, I'll tell you this: if you start an organization like that, I don't think anybody is going to bother you. They are not going to invade your property to leave threatening symbols or put a tracking device on your truck or trespass to take photographs. And they're not going to hang on your every word and get multiple people to pile on responses. That's because you are not going to be a threat. I'm sure they would rather have you than me any day.
What specifically exonerates Oswald? His alibi specifically exonerates him. And if you can't establish his alibi, then the most you are ever going to do is move him from Hell to Purgatory. You are never going to vindicate him.
There are other organizations that dispute the official story of the JFK assassination: CTKA, COPA, ROKC. But nobody bothers those people. There are no hate sites set up against them. And that's because there is no "O" for "Oswald" in any of their names.