James, one thing at a time. If the engine is running, the ignition coil is working. If the engine stops running, the coil may be opening and failing to produce spark, or, as you guessed, the battery may be flat because the generator isn't charging. If you immediately put another battery into the machine and it starts right back up, the coil isn't the problem, but if it's losing the magic juice it's life is probably limited. Pick up a spare the nest time you are at the auto store.
You do have the oldest of the systems I mentioned, something I sort of suspected. Is this a 6V or 12V system? In many ways, this is the most robust of the charging systems available for that engine. But as old as it is, it's not unreasonable to expect failures.
I don't know what caused the system to stop charging, but I got the impression from your earlier comments that it was working OK until something was disturbed. If so, it is less likely that some part of the system failed and more likely that a connection was improperly made, or missed.
Does your ammeter point to the left (discharge) when the engine is running and straight in the middle when switched OFF? If so, it's OK and the problem lies elsewhere. With the ignition switch ON but engine not running (and with a good battery) do you have 12V at the BAT terminal of the voltage regulator? You should. If not, trace that circuit back to the battery to see where the problem lies. That lead should go through the ammeter to the battery. That's where your charging current comes from. The regulator should have two more connections, FIELD and ARM (armature). These should be connected to the appropriate terminals of the generator. Believe it or not, that's about all there is to the connections, unless there's ground strap or wire in there somewhere.
Try cleaning the connections one by one. Remove the fastener (screw or nut) and clean away any rust or corrosion so that a good metal to metal connection is made. One possible quick fix is to try repolarizing the generator. With the engine not running, and using a piece of scrap wire, quickly connect the BAT terminal of the regulator to the FLD terminal. Just a quick touch should do it, not a permanent connection. If the generator now charges, you are done. If not, you have to dig deeper. The generator brushes may be worn, sticking, or making poor contact - or a combination of the three. The regulator contacts may be burned or corroded. If they're not to bad, they may be able to be filed clean. If none of these are successful at getting the system to charge, I suggest you take it to a dependable local starter rebuild shop. Old as that thing is, it has earned a good rebuild.
But be aware - if you remove the generator, you will have to retime it upon installation to get the distributor in correct orientation. If you don't have the timing instructions and the know-how, I suggest leaving it alone until you do.
Too bad you're not closer. I think I have on of those generator/distributor units on a shelf somewhere.
Steve
EDIT: Looking at your picture again, it appears that there is no connection to the terminals on the regulator. There's that blue wire that disappears somewhere behind the generator. I assume that is going to the ignition coil? I can't tell much about your wiring in the detail pic of the back of the gauge panel. The short fat wire from the start switch should go to one side of the ammeter. That's all that should be connected there. At the other side of the ammeter, there should be two wires, one to the ignition switch and the other to the BAT terminal of the regulator. On the other side of the ignition switch, that wire goes to the ignition resistor (if used) and the coil.