It's more that fallout shelters are basically useless even if you manage to get yourself inside before the detonation.

Basically, you would run out of water, and even if you have clean water you would still run out of food, even if you survived the strike and the fallout. We cannot actually live without major transportation hubs to deliver food etc., which by necessity are located in or near the major urban areas that would be destroyed in a nuclear war. That, more than anything else, is why the US and USSR ultimately abandoned "civil defense" and focused on "deterrence."