The reason they don't have piston kits is because they're selling OEM Honda parts, which all came separately from Honda as they didn't sell kits either. It's more of an aftermarket thing when it comes to hard parts like pistons and rings, since it's all aftermarket they gather all the parts you'd typically use. One exception back when these bikes were sold and worked on daily was gasket sets, Honda sold top and bottom end sets as well as individual gaskets. Notice that even aftermarket doesn't sell valve sets with stem seals and new springs either.
First, the obvious to me right away is this engine should be taken completely apart. It looks inside like the oil wasn't changed for 10,000 to 15,000 miles, nasty dark sludgy look and darker than most I've seen.
As to the transmission, this might simply be the common misconception about constant-mesh transmissions. Even if both shafts are turning because of oil friction, sometimes they won't shift more than one gear away from neutral because of the design. the engagement dogs on the sides of the gears have to align in order for the shift drum to move the shift fork for that gear and engage into the one desired for that speed (1st, 2nd, 3rd, etc). If you're not familiar with these, this will help. Ignore the obvious differences in the shift shaft and mechanism in the first part, the later displays show the engagement dogs and how they have to interact with each other.