That narrows your "doesn't do anything with the key" problem down to interlock switches, wiring or starter solenoid. All we did was bypass all of those and determined that the engine will indeed "try" to crank when supplied with battery power. To fix that, you've got to find the bad item in the "start" loop.
But, the video shows a very sluggish cranking effort, not nearly enough to reliably start the engine. The causes of this can be numerous, ranging from a bad ground connection to incorrect valve clearances. If you have a voltmeter, you can check the ground connections, circuits and switches in the interlock circuits, one at a time, until you find one that isn't working properly. With that taken care of, the first thing I would check the cheap stuff first--make sure the starter mounting bolts are tight, and check the valve clearances to make sure they are within spec. Most air cooled gas engines have a compression release to aid in starting and can be very sensitive to valve clearance misadjustments.
If you understand what I'm talking about, proceed. If you don't, you might want to get someone to set the valves for you.
If that doesn't get it cranking fast enough to start, then you probably have a bad starter.