Stator can be a concern, yes. The problem isn't so much the stator as the regulator/rectifier.
LI-ION variants are sensitive to voltage levels, but more at the high end and less at the low end (though both are bad). Over voltage will kill a LI-ION in short order, so you should definitely replace the R/R you have with a more modern option. The one on our site works well for LI-ION.
For the low end, the main concern is running the battery down too low. Too low on LI-ION means a dead battery. Permanently.
The problem with the Honda twins of the 60s and 70s is that aside from having voltage levels too high while they're running, they also don't put out enough power when they're idling. You don't usually start making usable wattage until 3000 RPM (with the stock setup) or closer to 2000 RPM with our R/R.
You have a few options. LED swap all around will definitely help. The move to a Rick's Hotshot stator will also help. The cheapest and best option is just to avoid idling it for long periods of time. Avoid city riding and traffic jams, if you can help it. Don't leave it warming up in your driveway for 15 minutes, etc.
For the LED headlight, it's very unlikely that you'll find a sealed beam version, but if you do, please let us all know. You'll probably have better luck buying a new bucket and a separate bulb (which we also have for sale on our site).