Does the generator charge the battery? I have no idea. Should it? Are you asking me to check this? I think I will add voltmeter to my shopping list along with my battery charger. I have a voltmeter with a light and it if there is a current the light goes off. Will that work? I am pretty sure the issue is when the tractor is running.
The light bulb is sort of a "poor man's voltmeter" that helps with telling you if you're getting enough juice to light the bulb, anywhere from ~11~14 volts, unfortunately it won't tell you how many volts exactly which is what you need to know for troubleshooting.
You are now in the realm of "troubleshooting," key word being Trouble, and as Chopperhed said a voltmeter really helps a lot in this difficult arena. Without knowing whether your battery is being charged while running you will just be guessing, perhaps pulling off parts and wires, maybe buying new parts blind, and not knowing exactly where the problem is. Now that winter is here I would be trying to figure out as quickly and directly as possible where problems are and a voltmeter is the best way to eliminate electrical guesswork.
A cheap analog voltmeter, the kind with a needle, is probably under 20 bucks at a hardware store. They are small and hard to read accurately, especially with the engine running and things bouncing around.
A cheap digital voltmeter will probably cost a little more at a place like Car Quest, etc, but is much easier to read and will tell you exactly how many volts you are getting. A good digital voltmeter costs even more but will pay for itself in terms of not letting you down when you really need it over the years.
For example, if you charge your battery on a charger then start the tractor, a light bulb touched to the battery terminals will glow bright, but you won't know exactly how many volts it has and if it is being charged by the generator or discharged by the ignition.
If you take a reading on the same freshly recharged battery with an analog meter, you will have to squint at the needle and try to guess which lines it is falling between while things are bouncing around. 12-13? 13-14? And where is it reading when the engine isn't running?
Now put a digital voltmeter on the battery with the engine off. If the battery is freshly charged it might give you a reading of 12.90 volts. Now start the tractor and take a reading. If it still says 12.90 volts or less, you know for sure that no current is reaching the battery. You can then start backtracking the electrical circuit with the voltmeter leads, to determine exactly where the problem is. For example, as MH81 shows, you can backtrack to the regulator to see if that is working. Or all the way to the starter/generator terminals to see if it is generating in the first place. If the battery reads 13 volts or higher while running, you know your charging system is in order. If it reads over 14.5 then you have an overcharging issue but that doesn't sound like your problem. Just trying to say, a decent voltmeter takes all the guesswork out of it. It will also make it much easier to get help online if you can say what a voltmeter is telling you and where. Just my .02, good luck
p.s. just as an example, i bought a small Fluke digital voltmeter about 18 years ago, it was made in the US and cost around $100. I have no idea what they cost any more as that meter will probably outlast me. And not endorsing any brand as most stuff is now made cheaply overseas. Before that I was going through $10 analog voltmeters about every two years, they were always hard to read and crapping out at the worst time. Before that I had a light bulb on a wire which always had me guessing. Not telling you what you should do, just telling what works for me. good luck