If the spark isn't hot enough, it won't fire. New plug, new carb, fresh fuel, new battery; the magneto is the only thing left in the mix that wasn't changed. When the new battery was added, it had enough juice to create a hot enough spark to fire. Once fired for a moment, it created enough heat to continue running as the battery level went down, and once shut off and cooled, it no longer had the fresh battery charge to create a hot enough spark. Giving the battery a boost may let it fire up again, but this really sounds like a weak spark. Spark+Fuel don't always make fire. The spark needs to be hot enough to do the job. Also, high octane fuel needs a hotter spark than low octane fuel, so this could have added to the condition depending on the fuel that was in it, and the fresh fuel that was added.
