I give six reasons why Aristotelian eudaimonia is not well captured by "flourishing". These deficiencies are so great that, in my view, that rendering should be avoided--it is better simply to transliterate or to re-educate as to one possible (and perhaps the preferable) use of "happiness".
(Understand always below: eudaimonia as used by Aristotle.)
1. Eudaimonia is an absolute term, "flourishing" is relative. There is a single standard of eudaimonia in the universe, and what human beings can achieve counts as eudaimonia to the extent that they approximate to this. In contrast, "flourishing" is species-relative: each species flourishes, if at all, in its own way, without any species serving as the standard for all others.
Broadie in a sense agrees: "it would be a mistake to propose "flourishing" as the preferred translation for eudaimonia in general. The world excellently captures the narrower idea of human eudaimonia, but -- as Aristotle does not allow us to forget -- eudaimonia was ascribed to the gods as well as to the best and most successful humans." But it's too weak to say that Aristotle merely "ascribes" eudaimonia to the gods as well: he takes the gods to provide the true case of it.2. Eudaimonia has a double function, which "flourishing" does not admit. Sometimes it is said that Aristotle 'equivocates' in his use of eudaimonia: he uses it to characterize both (i) a life as a whole and (ii) a distinct sort of activity (sc. the actualization of the virtue of sophia). But it's probably better to hold that the word can intelligibly be used in these related senses. Note that we do have words in English that function in a similar way. Think of "success": it's intelligible to say both that someone's life as a whole is a "success", and that some of his distinct achievements count as a "success". And we would naturally say that his life was a success because of the presence of success within it. But "flourishing" cannot coherently be used in this way.
3. "Flourishing" does not match correctly the connotations of eudaimonia deriving from its etymology. Eudaimonia is variously taken to signify in its origin either (i) "favored by a divinity" or (ii) "having a favored divinity (sc. spirit within)". (i) connotes that someone has been given something as a gift, or has been favored by good fortune, whereas flourishing, even if it suggests fragility, as Broadie says, connotes neither of these things. (ii) connotes that something special within us or somehow set apart is well off. This is connected with the notion of an absolute standard for eudaimonia, which, as we have seen, "flourishing" fails to capture.
4. "Flourishing" has connotations which would be inaccurately imputed to eudaimonia. Because it is involves basically a vegetable metaphor, "flourishing" suggests the manifestation and unfolding of something already contained within. (Note that it does not, itself, even suggest fruitfulness.) But Aristotle does not conceive of eudaimonia in this way (it's rather something like an achievement), nor is eudaimonia, pace Irwin, a matter of 'self-actualization' or 'self-realization'.
5. We cannot "flourish" except at the height of our physical powers, but eudaimonia has no such restriction. When Thomas Reid, 75 years old and nearly enfeebled, retired from lecturing at the University of Glasgow and had leisure at last to write out his lectures (which became his treatises on the Intellectual Powers and Active Powers), we might have been justified in ascribing eudaimonia to him, but it would have been an abuse of English to say that he was flourishing.
6. Eudaimonia is related to "blessedness", which cannot be construed as "flourishing". We must look to the adjectives as well as the substantives: Aristotle frequently pairs eudaimon with makarios, as if these were equivalent or very similar, and yet I should think it is clear that makarios does not mean "flourishing".
In any other case of a proposed rendering into English of a Greek term, if the rendering were similarly beset by difficulties of comparable seriousness, we would either reject it entirely or regard it as extremely unsatisfactory. That is why I find Broadie's talk of "the great Aristotelian idea of human flourishing" to be so puzzling--since "flourishing" is not Aristotle's idea and doesn't capture his idea.
But why do people so commonly say that eudaimonia is best rendered as "flourishing"? Not, I think, because "flourishing" suggests vulnerability (which Broadie stresses), but rather because people look for a term that indicates something which meets objective standards, is common for a species, and which is non-transient.