I share your dislike for "Anonymous" display names, Jack, especially since I'm not at all sure that there are any checks in the system to avoid duplication, so it could well be that we have dozens of "Anonymous Smith"s or "Anonymous Anonymous"es on the system. It would be nice to know, when interacting with an Anonymous Person, whether it's the same Anonymous Person with whom I have had previous interactions or not.
(I do know that there's a member who shares the same first and last name as one of my cousins and is fairly active on G2G, but is not, in fact, my cousin, who is on WikiTree, but has never, to my knowledge, posted on G2G, so it's kind of weird to see her name keep popping up when it's not actually her. I imagine that we'll have more of that happening as more people join the site, so it might be nice to be able to set my display name in G2G to something like "Greg (Grga) Slade" if any of my many co-named people show up here.)
Nonetheless, I recognise that there are plenty of people who really don't want their names out there in public for a variety of reasons (especially since G2G is searchable through search engines, which is good for cousin bait reasons, and I've had one old friend find me through my presence here after I quit Facebook a few years ago, but not good for people trying to avoid abusive exes or whomever). So while I would be wholeheartedly be in favour of encouraging the use of real names, I would be very uncomfortable with making it a policy.
I also think that it might help with the whole "Anonymous Anonymous" issue if people could pick nicknames. While I would personally prefer to address people by their real names (because starting a message with "Dear Anonymous" is just weird), at least letting people use nicknames like "Conan the Librarian" or "Anne the Ancestor Hunter" or whatever would be a little less weird.