As veiwed from the side-car of the Ural

Sometimes a step backward is really a step in the right direction.
Tony and I had been piddling around with the Honda some for a couple of days without a lot of success. To be honest it was kind of a half hearted piddle, until yesterday.
For a few days it was: try to start it, it catches briefly and dies, and we are looking at on-line trouble shooting guides to find the symptoms...fires up and then just quits. Gradually it gets to sounding a progressively worse, until it gets to the point where sometimes it starts to sound like the starter clutch isn't actually engaging all the time. Then Thursday we began trying to get it started via the kick starter, forsaking the convenience of the electric starter to avoid hearing the sound. The results were the same until...it just locked up. Totally locked up.
In my mind I was thinking that it was the same old problem with the alternator/starter clutch.
So yesterday I pulled the left cover and sure enough... The new starter sprocket retaining plate was bent, the sprocket had just about totally disengaged from the starter clutch, and the starter chain was bound up against the side of the crank-case housing and tight as a banjo string (I used to have a banjo so I know how tight the strings are and that's not just a trite expression I'm throwing around loosely.) For some reason one of the screws attaching the starter clutch to the alternator rotor had backed out (I lock-tited them) and forced the sprocket out of position. This was a problem with this bike before I got it, and after seeing what happened this time I'm figuring this is how the kick-starter shaft got broken in the first place.
I don't know why these screws won't stay tight, I just know they won't and I may be at the point that I don't care.
We take all that mess out, (by the way, all the parts I've been ordering and waiting on for the last month were in that little sub-assembly), readjust the idle screws on the carbs back to the point that someone recommended either here or on facebook, and kick her over and...whoa! Fires right up and is idling smoothly at 1400 rpm! We repeat this several times just to be sure it wasn't a fluke, and it fires up everytime with one or two kicks.
So we spend an hour or so checking and adjusting the brakes and adjusting the clutch (is the sweet-spot on the clutch adjustment really that hard to hit?) before deciding that we've reached the point that there's nothing left to do but to do it.
Tony follows in the Ural and Desiree records the event on her i-phone from the side car.

And now some questions...
At the end it died. The ride was just over a mile. I was just going around the block so luckily when it died we were only about 200 yards from the house.
It seemed to be running pretty well initially. Plenty of power, plenty of pick-up, but after about 3/4 mile it strated coughing and missing pretty bad.
When it stopped the battery was dead. No horn, no nothing.
We pushed it up the street a ways and when we got almost back Tony tried it and the horn was working and it fired up (less than 5 minutes elapsed.)
So, what to look for now? Not charging? Dead short? Where should I start looking and what testing can be done?