For my own experience "charged the battery" is almost always the failure point. Batteries go bad and won't take a charge. I'd use a multimeter to check that the battery voltage is actually there. If not, replace it. This is especially true for a battery that goes a full season or two without being used at all - especially if your Willys happens to have a small closed circuit somewhere. Otherwise you get to debug some wiring.
One unfortunate thing I've experienced TWICE in the last couple of months with trailer wiring is that water and corrosion can be drawn WAY up inside of a wire from an exposed end, and rot it for several feet, making it unable to carry a current. So just because a wire is in place and "looks solid", doesn't mean much. If you don't find any obvious breaks in the circuit to the starter then you might need to use your multimeter to do continuity and/or voltage testst across each length of wire - as others have suggested.
But, based on your description it sounds like you could have both an electric and fuel problem, though it is possible to get all of these symptoms just by electric.
Once electric is out of the way, drain the gas from the carb bowl into a clear jar and look for a layer of water on the bottom. The water will get pulled into the engine first, and depending on how much is there (hopefully none) will cause run / stall / run / stall situations. If there is water in the bowl, there is water in the tank, lines, filter, and pump, so the whole thing needs to be well drained. Water sinks in fuel so it will go to the lowest possible spot, and wait for you to add fuel on top of it. It's "neat"
