Slow progress. I've got other things demanding my time (mostly a newborn baby), so not much time to fiddle with the bike at all. I didn't expect a project when buying this bike, but oh well. Life goes on. I'll make it right eventually.
So far I can say a few things. There were some small issues in the selector box that I rectified. The selector wheel/shaft was not moving smoothly and was binding slightly in the bushing it rides in. I've also tried turning the selector drum directly via the stub end that comes out of the transmission and selecting gears while just turning the back wheel; every gear position, including 4th, can be selected positively and cleanly and remain engaged by directly turning the drum. Finally, I can reiterate that the stock linkage, upstream of the selector box, on these '76/'77 SSs is objectively terrible. The issue is not the crossover shaft itself or the number of connections in the linkage, but rather the specific connection between the shift lever and the actuator arm attached to the crossover rod. It's a sloppy sliding pin/slot interface with a lot of friction and freeplay; it's an inefficient and imprecise design when the parts are new, it only gets worse when the parts are worn. It's hard to appreciate how bad this detail of the linkage is unless you've seen the specific execution on one of these crossover shift Super Sports in person. Anyway, this sloppy connection between the shift lever and the rest of the linkage is such a fundamental part of whether or not a solid and accurate input is made to the selector box (and subsequently the shift drum) that I don't think it is worth proceeding until I address it. Converting to RH shift would be one idea as others have suggested, but I don't believe it needs to be that drastic; the various parts of the linkage just need to be properly connected with real rotating joints (ball end heim joints). So, I'll be doing some fabrication so that I can fit rearset pegs/levers and connect the shift lever to the crossover shaft via a proper linkage rod with balljoints at either end (similar to the stock '78+ SS peg/lever arrangement). Once I'm sure the linkage is precise and I know the transmission is getting good, accurate inputs, then I'll reassess the shifting on the road.
@wdietz186 Your description of the root grooves of the gear teeth being machined right down to the engagement dogs is clear and makes a lot of sense as a weak point, but I only see that happening on 5th gear (on the layshaft). On all of the other gears in the box, it seems there is plenty of radial clearance between the outer diameter of the engagement dogs and the root of the gear teeth. If there is a specific structural weakness with the dogs on 3rd or 4th gears, what is it? Having not seen the parts in person, I certainly could be missing something, but by looking at photos of parts (individual gears and complete gearbox assemblies), I can't see anything that stands out about the design or machining of 3rd/4th that looks uniquely problematic.
-R