From your video, the stove certainly looks well made and I'm impressed with it's clean burning.
Is there possibly a waiting time programmed in so the Peltier device has a chance to heat up and generate, so as not to deplete the battery with the fan before charging an outboard device? If the green diode is lit, it would seem to indicate "ready".
Volt/Ohm (multi-meter) meters are cheap now days and would help troubleshoot your stove (and many other things!).
For your test, your USB light should be enough.

If I were testing the stove, I'd get a fire going and keep it going strong for 20 min. If it's not charging by then, you probably have an electronics problem. Warranty?