'...one of those early Unix desktops "is still alive"...'
no, it's not. NextStep was not "Unix" and Mach/BSD was not a desktop.
Actually, OpenStep was licensed UNIX. So, legally it was UNIX. But that's just legal pedantry. NS and OS were unix in all the ways that mattered. As is MacOS. They ran all the unix tools (some version or another). They did TCP/IP. Hell, they even ran X11 if you wanted to.
My father once asked me if linux was unix. Almost the same answer - legally it was not, but in every way that mattered it was. What's more, it was going to win (the unix wars).
In a sad way, MacOS and linux are the tied losers of the desktop world. I don't even know offhand what "legally branded UNIX" there is any more. Does Oracle sell some flavor of branded UNIX? You can argue that linux is more unix than MacOS - and that's fine - I'm not going to argue it. But I use MacOS because I like unix and I don't like windows and I do like having a nice desktop experience.