It looks like states are focused on not having vaccine passports in government entities not as a whole. Many also say that healthcare and traveling are exemptions and they let local businesses do their thing.
https://www.usnews.com/news/best-states/articles/which-states-have-banned-vaccine-passports. Arizona "Ducey added in a statement, "While we strongly recommend all Arizonans get the COVID-19 vaccine, it's not mandated in our state – and it never will be. Vaccination is up to each individual, not the government."
If there weren't other variants, maybe I can see it working in the short term but for now it looks like a short-term fix for a possible long-term problem.