benhogan... If you want to experiment, you could 'flip' the lever that engages the dog clutch instead of cutting a shim like the u-tube video shows. Even drill holes 180 off for th springs, etc. If you don't want to go that far, try trading the hi-low shift lever for the engage lever. It would let you test...
FWIW, the hi-low shaft shown in the video seems to have a LOT of play in the housing, so there is a bit of wear letting the lever flop around, or it was designed to be loosey-goosey from the get go. I ain't a Honda engineer, so can't say. Cannot also say about the pinion gear shaft as it seems to be supported by one bearing, which would allow the gear to get out of position relative to the spider gears as things wear. Belt tension pulling in one direction all the time would lead to bad things. Guess I'm saying that as simple as the gearbox is, it was designed to a price. Maybe I'm too paranoid, and there's only fractional hp being transferred, but having a shaft with one bearing is not good design in my view.
I'd bet the pulley on the mower you have is allowing the belt to slip as it warms up from use. New may fit properly if the bearings are good and tight, but that design shouts about the belt loosening under use if the bearings have any slop at all.
Wiggle the input pulley to check for play...
tom