The argument that recordings from the 1960's would not be played in the 2060's because people nowadays don't listen to 100 year old wax cylinder recordings holds no water. The reason why so few songs recorded on wax cylinders are played today is that the technical quality of them is laughably terrible to our ears. The quality difference between the old wax cylinder and wax disc recordings and modern recordings made on tape and transferred to vinyl is several orders of magnitude. The quality difference between a professional studio recording of the 1960's and a modern professional studio recording is a much, much smaller interval. Yes, modern recordings are a little better technically, but not enough so to make the 60's recordings unlistenable. No one who argues that people will listen to music format programming on AM radio stations has a leg to stand on claiming that 1960's era recordings aren't good sounding enough for people to listen to them.
So then you get to the other specious argument about older recordings, which is that people only listen to older recordings from their own youth to bring back nostalgic memories. That's such a crock of poppycock that there's no need to disprove or refute it. Anyone with a lick of sense knows it a bogus argument. Nostalgia might be one factor, but it's not the only factor. It's not even the main factor.
Any station today that plays Michael Buble or Harry Connick, Jr. would find that the people who like those two would also like Sinatra's music if it were played. There are really two things at play here. One issue is whether audiences in the 2060's would still like the music of the 1960's if they heard it played. The other issue is whether or not the suits controlling what gets played would give audiences the opportunity to hear music from the 1960's in the 2060's. That's a whole different question. I predict that by the 2060's OTA broadcast radio will be a memory, and people will all get to pick whatever they want to year without some arrogant suit controlling access to music.
There's plenty of music that "transcends the ages". But there's not much that transcends the ham-fisted censors of modern music who claim that "testing" determines what they play. That's why OTA radio as we've known it for the past half century is dying.