His impulsive actions are overblown in the grand scheme of things. Had he lived, Oswald strikes me as someone who would have denied the crime right up to the electric chair.
If he truly wanted to become a historic figure then I don't think so. In any case, I think he'd have to give answers to the questions about, among other things, where he was at the time of the assassination and what he did afterwards. Or in terms of physical evidence: could he publicly deny owning that rifle? And all of this nonsense (to me) about him being a CIA agent or asset would be exposed for the folly it is. Then we have the lesser items: Mexico City, et cetera.
At some point his alibi, his attempts to answer these and other questions, would collapse on its own inconsistencies and contradictions. Other than the hard core types - the two Oswalds believers, he was a victim of MK-Ultra, the Sirhan types - none of us would be here.