QuoteTo rely on their existence because everyone believes in them doesn't seem consistent.
I have my own questions on this topic, DaveT , but I have come to an understanding that may be helpful on this one.
Here is Bailey's translation of the text from the Letter to Menoikeus:
Quote[123] The things which I used unceasingly to commend to you, these do and practice, considering them to be the first principles of the good life. First of all believe that god is a being immortal and blessed, even as the common idea of a god is engraved on men’s minds, and do not assign to him anything alien to his immortality or ill-suited to his blessedness: but believe about him everything that can uphold his blessedness and immortality. For gods there are, since the knowledge of them is by clear vision. But they are not such as the many believe them to be: for indeed they do not consistently represent them as they believe them to be. And the impious man is not he who popularly denies the gods of the many, but he who attaches to the gods the beliefs of the many.
Regarding the sentence I've underlined, I no longer read this as Epicurus offering evidence of the existence of the gods. Rather, I think he is offering a definition of a god. This is in line with the pedagogy described in this passage in the Letter to Herodotus:
QuoteFirst of all, Herodotus, we must grasp the ideas attached to words, in order that we may be able to refer to them and so to judge the inferences of opinion or problems of investigation or reflection, so that we may not either leave everything uncertain and go on explaining to infinity or use words devoid of meaning.
So when he says in Menoikeus that a god is 'a being blessed and incorruptible', he is saying that those are the ideas attached to the word god. It's not important or evidentiary that 'everyone believes in gods'. It is important (at least to Epicurus) that people commonly have an idea about the thing said to be represented by the word god.
The actual evidence comes in the next sentence in Menoikeus: we know the gods exist because of 'clear vision', ἐναργὴς. They are manifest. The 'true impression', or preconception, that is, the prolepsis of the gods, seems to derive from this clear vision.
TL:DR; From the point of view of the individual, other people are not involved in helping to determine the existence of the gods. They are only useful insofar as they ratify our grasp of the meaning of the word god.