I think you have agreed that "God did it" is inadequate, because this is an argument from ignorance, which you reject on philosophical grounds.
In my earlier post, I suggested that repeatable, verifiable evidence of a supernatural/unnatural force would make this force natural, by definition, and thus dethrone gods.
There is another notion, namely, that gods or God can intervene willy-nilly changing everything according to their whim and omnipotence. It seems to me that this notion would destroy every concept in objective reality, methological determinism, and usual common sense.
But this view of an invisible, supernatural deity intervening randomly from time to time according to pleasures, and who is impervious to cause and effect, seems impossible of scientific examination.
Because the word "supernatural" is scientifically and objectively arbitrary (i.e. subjective), it is useless except for the purpose of politicizing free inquiry. Personally I attach no value to the word because it, like the word "natural," could apply wholly to all of existence. It is a non-falsifiable qualifier from the get go.
If one wishes to take up the position that "science cannot investigate the supernatural," then let him know he has taken up a philosophy that will color each and every hypothesis as well as the interpretation of the evidence. This position, useful as it may prove both intellectually and practically, is not subject to scientific validation.