The Epicurean acceptance of the existence of the gods didn't just rely on abstract theories but also on the idea that the existence of the gods is an empirically verifiable fact because of visions or images of them streaming into our mind. So Epicurus is basically telling us that we know gods exist because many people have actually seen them.
I don't think that (which is the foundation for the rest of the post) is correct at all. The weight of the evidence as I understand the sources is that the images of the gods are not perceived by the eyes, but by the mind.
As DeWitt begins his chapter on the Knowledge of the Gods:
QuoteKNOWLEDGE OF THE GODS
An inveterate tendency to classify Epicurus as an empiricist has resulted in the conclusion that according to his thinking knowledge of the gods comes by vision. The absurdity of this view will become clear as abundant items of evidence are assembled against it.
According to these evidences the sources of knowledge are multiple. The Prolepsis apprises men of the blissfulness and incorruptibility of the gods. The Feelings, that is, fears and worries, serve to inform the individual of the true nature of the divine through the distress that follows upon "false opinions." Reason, by deductive inferences from the Twelve Elementary Principles, informs men of the existence of gods, of their corporeal nature, their number, their gradation in kind and their abode. By the method of analogy, that is, progression from similars to similars, reason also produces confirmatory evidence concerning their form, by a chain argument concerning their nature, and by a disjunctive syllogism concerning the kind of life they lead.
And I would in no way or form equate the Epicurean view of the gods with mass hallucinations.