You are arguing, if I understand you aright, (1) that the gods don't want their existence to be widely known but (2) that encounters with the gods, dramatic enough to demand extraordinary explanations if they aren't real, are commonplace.

This seems like a curious combination of claims. Could you say a little about why you don't find their conjunction wildly implausible? (Or, if the real problem is that I've badly misunderstood you, correct my misunderstanding?)