Charging problems

Sounds like it, but there are some tests to perform first to rule out other causes. You'll need a test light or a digital voltmeter. If you don't have one, you can find them for less than ten dollars at Harbor Freight Tools, Walmart, any hardware store, or any auto parts store. Don't buy a meter with all kinds of features you'll never use, and if you buy a test light, don't buy a fancy one with colored LED lights. Those are fine for other types of problems, but they can give misleading results here. You want the cheaper one with a regular 12-volt incandescent light bulb inside.

If you don't know how to use the meter or test light, these articles will help:

https://www.2carpros.com/articles/how-to-use-a-test-light-circuit-tester

https://www.2carpros.com/articles/how-to-use-a-voltmeter

https://www.2carpros.com/articles/how-to-check-a-car-alternator

If you need help setting the voltmeter or how to read it, I can help with that. Start by measuring the battery's voltage with the engine not running. It should be close to 12.6 volts. 12.2 volts indicates it is good but fully-discharged. It should be charged at a slow rate for a couple of hours with a small portable battery charger. If you find it's around 11.0 volts or less, it has a dead cell and must be replaced.

Next, measure the battery's voltage again with the engine running. It must be between 13.75 and 14.75 volts. If it is low, the charging system is indeed dead.

Move the probe of the test light or meter to the large output stud on the back of the generator. You should find the same voltage there as you find at the battery, at all times. Be sure to check that with the engine off and with it running. If the two are places different, we'll have to look for the very large fuse in that circuit.

Watch the instrument cluster when you turn the ignition switch to "run". The "Battery" warning light must turn on. If it does not, we have to look at that circuit. Does that light turn off when the engine starts running? Let me know what you find up to this point.

Look for the three-wire plug on the side / back of the generator. You'll need to back-probe through the rubber seals alongside each wire. Check the voltage on the yellow wire. You should find full battery voltage all the time, or the test light should be bright. Of less importance, check the voltage on the white / black wire. That one will have exactly half of battery voltage, roughly 7 volts, or a dim test light, when the charging system is working. If you find 0 volts, that is proof the system is not charging the battery.