I agree "happiness" is not an adequate translation ofευδαιμονία eudaimonia. However, I also do not like "flourishing" as a translation of eudaimonia.
Flourishing is primarily defined as (Merriam-Webster) "marked by vigorous and healthy growth" (a flourishing garden); "very active and successful."
I could maybe accept it if one goes with the sense of "successful" as in " having attained a desired end or state of good fortune" but I don't normally get that sense from "flourishing." To me, that definition is better attached to "well-being" "the state of doing well especially in relation to one's happiness or success"
Well-being - Wikipedia
en.wikipedia.org
I also like "well-being" because it wordplays off of eudaimonia itself: eu "well" + daimon "a being/god/deity".