I got this thing to shift, finally.
So, for the sake of anyone with a fucked up JLO TS50 motor who stumbles upon this thread, I'll tell you how.
The second gear slides up and down a shaft, constantly spinning on the shaft when in 1st and third, and locks into tapered splines on the shaft when shifted into second.
As it gets old and and the splines wear, it binds up and jams, and won't slide over them anymore.
So I grinded down the splines on the shaft to make it all sloppy and rounded, so the sliding gear wouldn't catch. So instead of looking like a wedge, it looks like a hump.
I only test rode it around the block 1 1/2 times (because I forgot I drained the tank) but it shifted through all three gears without binding.
How long will it last before total failure? Who knows. But if the second gear fails completely, at least it'll have 1st and 3rd. Probably.
Also, if anyone needs any help rebuilding a JLO TS50, I can do it almost blindfolded now. I think that's the fifth time I've rebuilt it.
And the bike might be going up for sale soon. Not cheap.