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Here's a possible explanation that you might not have thought of....

Oswald's Carcano rifle, per the FBI's firearms expert Robert Frazier, fired bullets high and to the right when using the 4-power scope. [See Frazier's Warren Commission testimony, at 3 H 404-405.]

If this was also the case when Oswald was shooting at JFK on November 22, and Oswald for some reason forgot this quirk about his scope when he squeezed off his first shot that day (if he used the scope at all, which is also debatable, of course), that could be at least a partial explanation as to why his first shot missed and (possibly) struck the nearby oak tree, a tree that would have been to Oswald's RIGHT if he was aiming a little to the tree's left through the scope just as JFK's car was nearing it from LHO's point-of-view (as illustrated in CE888 and CE875).

The Bottom Line (concerning the "missed shot"):

Nobody can know for certain what happened to that bullet. And nobody can know for certain whether the "oak tree" theory is accurate or not.

But, given the overall evidence (which certainly indicates that three shots and only three shots were fired during the assassination, with all three of those shots coming from Oswald's Sniper's Nest in the TSBD and from Oswald's Mannlicher-Carcano rifle, with two of those three bullets striking the victims in the limousine), I think the best guess re: the one missed shot is that that bullet did, indeed, hit the oak tree (which is a tree that was located to the RIGHT of Oswald at the time he fired that shot at approx. Z160, which fits in pretty well with a misaligned scope that might very well have been aiming "HIGH AND TO THE RIGHT" during the shooting, although that's another thing we'll never know for sure; it's quite possible that the scope became misaligned when Oswald dropped the gun amongst the boxes after the assassination).

More:
https://jfk-archives.blogspot.com/2010/07/missed-shot-controversy.html

As for LHO being "nutty"...that might be viewed as a plus. A nutty LHO's testimony might be taken with a grain of salt, and more-easily dismissed, providing plausible deniability.

Anyway, I am not sure LHO was regarded as all that nutty pre-JFKA.