What you're suggesting already exists. Nobody gets the smallbox vaccine anymore. People only get the rabies vaccine after exposure. There are a handful of vaccines you only need if you're traveling to certain international destinations [1]. The set of vaccines that everyone gets is already subset that everyone needs. And because of people expressing unfounded "skepticism" like this, we're getting outbreaks of diseases that were being kept at bay by the vaccines.

[1] https://www.cdc.gov/features/vaccines-travel/index.html