We used to use 85W90 gear oils, with molybdenum disulphide mixed in for the dry clutch H1R and H2R's, but moly lube shouldn't be used in a wet clutch street bike, 80W90 gear oils, YES, moly additive/lube, NO.
Oil "flow" is to the cup at the clutch end of the
output shaft, through the hollow shaft to the individual oiling holes for each
gear, then, down the gear and slung to the other gears on the input shaft. The
cup is filled by oil slung off the rear of the outer clutch hub.
The damage looks exactly like the oil just wasn't up to the job of lubing the trans. I've seen a lot of this, usually from too light a viscosity, or synthetics in use.
I've not seen a damaged transmission when using a
natural gear oil, but have seen very damaged ones that used "special
transmission synthetic gear oils" in them.
First thing I used to do when doing a new bike setup, or service, was to swap the trans drain plug for a magnetic one. Makes it a lot easier to see what is in the transmission fluid.
I've seen many 3rd gear shift forks damaged, no matter the oils, when the 3rd gear thrust washer was cupped from spinning, and side loaded the fork. When the no spin type tabbed washers were used, no issues with shift fork burning, even with the 21mm narrow forks.
Leo, the "cupping" I am referring to is, the washer between the gear and the circlip. The factory traced the problem to side loading of the gear against the stock early circlip that spun on the shaft with the gear. The washer would overheat, cup, and bend the circlip, the gear would try to ride over to one side on the shaft, which put a large side load on the fork, burning it. The fix is a thrust washer with tabs that fit into the grooves on the shaft, to stop the washer from spinning with the gear. Later Kawasakis, two and four strokes used/use them as well, in different dimensions. I believe mraxl has the service bulletins on these tabbed washers for the H series on his site.
You can use 4 of these tabbed washers on H series transmissions, only one washer won't clear the grooves on the shaft, as it sits in a position on the end of the splines, at their curvature, and there isn't room to clear the tabs all the way.
A better circlip is available from Yamaha, doesn't have the cut away at the ends of the circlip. Easiest to just go to a Yamaha dealer and match them up. Kawasaki uses this style in later models.