Whatever it does could just be assumed illusion. Any great tech and they're assumed an advanced civilization.
Indeed. It's a classic warning to the gullible: "Any sufficiently advanced technology is indistinguishable from magic."
Stage magicians can cut people in half and do all kinds of convincing stuff that stands up to the audience's scrutiny. James Randi, a professional magician, was a mainstay of CSICOP. He performed many of the tricks of faith healers, mediums, etc., for our meetings. He showed us how some of them are done and it's a combination of being a really good con man, having some really cool gadgets, and depending on the audience not to be science experts. (Walking on hot coals, for example, is easy so long as they're the right kind of wood.)
I can't conceive of anything something could do to convince me they're the one true god. Being god though, it could probably do something I cannot conceive of to make a strong argument in its favor.
Considering that David Copperfield makes buildings disappear, it would have to be quite a trick.
Did you all see the "Ardra" episode of "Star Trek: The Next Generation"? She convinced the people on a planet beleaguered by war that she was their god by using transporter technology, etc. She told them that if they would stop fighting and learn to work together, she would make them prosper, but if not they were doomed. Of course they stopped fighting and of course their civilization advanced. She came back several hundred years later to collect her fee, without even having done anything for them--except of course scaring them into ending their war!
It was an interesting dilemma. This is of course what the best and most honorable astrologers, fortunetellers, etc., do. They make you comfortable and speak to you in terms you accept as reasonable, and then they simply help you solve your own problems. A shrink I know said the best psychotherapist he ever met was an astrologer.