This is what I perceive to be the sequence of reasoning on this topic in the letter to Herodotus (clips from Bailey):
First, the atoms, which are eternal, do not possess any of the qualities that we consider to be in the nature of "concepts" or "universals," so "concepts" and "universals" cannot have permanent unchanging existence:
Next, even though the qualities of the combination of atoms (which includes all that we can experience directly in our universe) are not permanent and unchanging like the atoms themselves, we must not believe that they do NOT exist, OR that they have some kind of incorporeal existence. The things that we experience in our reality are real TO US (and this is the key to showing the insanity of nihilism):
And this is how "events" as arising from the nature and movement of the atoms is the explanation to which Thomas Jefferson referred. And this understanding is hugely important -- none of this is an "accident" in the way that you fail to look both ways before crossing a street and get run over by a bus in an "accidental" way. The structure of our universe as a series of "events" arising from the movement of the atoms, and is largely "deterministic" and understandable and predictable, except for the limited instances of "free will" (including the life of higher animals) that arise from the swerve of atoms and which are able to break through under limited circumstances.
But I fully understand why Bailey and others of his attitude would choose to use the word "accident" in these translations. They are essentially Platonic/idealist/theists themselves, they reject the views of Epicurus on this topic, they think that gods and ideal forms and universals are necessary to explain things, and so they prefer term which carries derogatory connotations ("accident" instead of "event" or even "conjuncts"). The 1743 edition has the preferred wording in my view.