I don't agree.
I know you're prolly referring to the WWIII, but it's never talked about much differently than WWII or WWI, if you don't account for the intensity of war changing. Like we consider WWI and WWII rather similar and for people living in a Utopia 200 years from now, a third WW about 100 years apart isn't that different.
I don't recall anyone ever saying the world ended during WWIII. Sure, First Contact, humanity has a definite post-apocalyptic vibe, but eh.
I guess you're technically not that wrong, but I wouldn't personally use that as I don't think anyone refers to the world having ended in the 21st century.
And the main setting is definitely utopian to the point chat we've never really gotten that many stories set on the Earth of that time, because it's kinda hard to write. Sure we can somehow explain how everything is equal but Picard still owns a massive vineyard with workers and is clearly well-off ans and it can't all just be from the merits hes gotten as an adult?