We can speak of four causes for the sake of clarity and limiting the scope of the discussion, but I don't think even Aristotle supposed there were only four. If the material cause of a table is the wood that it is made of, that answers to one of the proximate causes. But of course there are innumerable antecedent causes; the milling of the lumber follows the felling of the tree, which follows the growth of the tree, which follows the production of the acorn, and so on. But for Aristotle this can't go on forever; there cannot be an infinite regress of antecedent causes, so there must be an uncaused cause to start the chain.
The question (as raised by chatgtp) as to whether purpose-driven language should be used in biology is precisely one of the areas of contention. Lucretius seems to imply that it should not.
But in general I think that summary is ok. But we're not presenting Epicureanism to the professional philosophers of the world who already deeply understand these issues. We're presenting to other normal people like ourselves, and normal people use teleogical language, probably without knowing it, in areas it maybe shouldn't apply.