That's my point, and I think people in their 30s and 40s today are the same way. They know their parents music, and know it for that reason. But it's not something they connect with in a real way. So the real question is are they likely to be listening to a station that plays their parents music all the time? Probably not.
My parents refused most things contemporary. Mom likes big bands. Dad likes classical, but will listen to classic country when he doesn't think anyone is around.
I was exposed to all of these things as a kid. I appreciate them. But other than the occasional spin to the classical station when I'm at work, I have no urge to listen to them today. The stuff I listen to most is from the 70's, 80's, & 90's with a smattering of current stuff here and there. Many of the songs maligned in this thread are in my personal collection. I suspect that I am not unique for people my age.
The legacy of something from the 1960's doesn't mean too much to you when you were born in the 1970's. It just doesn't.