It's not that it's not possible to use a vertical shaft lawnmower engine. The problem is that it's not very practical. It's a heck of a lot easier to put a clutch on a horizontal shaft engine and line it up with the rear sprocket vs. having to put a bevel gear on the engine, put another bevel gear on a second shaft, put a clutch or tc or whatever on the second shaft (jackshaft basically) and then from there you have to get it to the wheels. Plus you'd have to find or have the gears made that will specifically fit your project.
I'm not sure I completely agree with what it says on that site about "disengaging the gears". It's just going to be more wear and tear on the gears engaging and disengaging them, especially since they wouldn't have any sort of lubricant.