I don't think it holds a lot of significance for any Americans born after the 50s either (how can we be nostalgic about something we never lived through?). So no need to feel excluded.
What's signifiacnt about it is the incredible kitsch factor of the cheesy old roadside attractions. It's corny, it's tasteless - it's 100% middle America. Isn't there an exceptionally Fallouty one of a giant statue of a guy in a radiation suit carrying a bomb or a missle - someone help me out here...
Also it's kind of symbolic of the idyllic innocence of the post war 50's when Eisenhower was president. Picture pulling into a drive-in restaurant in your Thunderbird (or Corvega), and being served a cheeseburger and chocolate shake by a waitress on rollerskates kind of era. Completely oblivious of the turbulence of the 60s looming - Race Riots, Viet Nam, Nixon etc.
Then the fact that Route 66 dried up after Eisenhower instituted our current highway system. When it became obselete, many of the towns and roadside businesses dried up and fell into disrepair - as did the road itself as no more funds were spent on maintenence. So Route 66 and that 50's lifestyle were kind of a victim of the natural evolution of things. What's left has become rusted, abandoned, or pretty seedy - very Fallout-esque.
Thematically a lot of it would fit Fallout like a glove