OK, well have you pulled the cylinder head off? You won’t know for sure until you do so, there is still the slight chance that something under the flywheel is jamming the engine even though nothing was obvious when you peeked under the flywheel.
I know how you feel, I have been disappointed more than once by a motor that was in nice cosmetic condition. You have paid a lot of money for repairs, but reported the gearcase was jumping out of gear. It may be that the linkage was just improperly adjusted, but who knows.
These Bigtwins as a rule, are good/reliable engines. You mention "chronic" problems with jumping out of gear. This issue surely was not chronic, and usually only occurs due to poor linkage adjustments. The wrist pin bearing issue Frank mentioned is unfortunate, and corrected in later models. You mention (at least I think you did) that your powerhead was completely rebuilt, but the crankcase screws appear undisturbed (not repainted also) in your pictures.
At this point, you have nothing to lose by pulling the head. I would recommend pulling the flywheel first, but you don’t have the proper tools for that. Perhaps the repair shop would be willing to help out by pulling that flywheel for you before pulling the head. Pulling the flywheel with the proper tools takes only minutes. Give them a call, you have nothing to lose. Have any local members replied to your inquiry?
Buying another used outboard may or may not be the answer, at least you know what you have with this engine, and it sounds like it is in real nice cosmetic condition. A serviceable 1958 35hp powerhead shouldn’t be that hard to find, which would get you through the rest of the summer. I’m sure we can find someone to help you out with tearing apart the old powerhead and evaluating it for rebuilding during the winter months.
I just had one more thought also. You mention it was "jumping out of gear" while running/skiing just before it failed. Perhaps what seemed like jumping out of gear was the engine jamming due to the needles coming out of the wrist pin bearing getting caught in the combustion chamber/ports. It just might be that there is nothing wrong with the gearcase after all.
I surely understand your frustration and hope something can be worked out to get you through the summer using this engine.
Don

Pull it apart! A clean survivor like that deserves a little CSI treatment to see what’s going on. Also, it is possible to swap 1958 powerhead guts into a ’57 powerhead. That would keep your motor going, give it the update, and allow you to freshen it up while you’re in there (hone, clean up carbon, etc.) ANY 50-60 year old powerhead has the potential to give you fits. Go through it right off the bat and you know what you have.