In Lithuania there is the "opportunity passport" which was designed to show you are safe. You can get it if you are fully vaccinated, have recovered (identified by a positive PCR test or have antibodies), or have had a negative PCR test in the last 48 hours. This allows you to show you are 'safe' without disclosing whether you have been vaccinated or not, so it solves the privacy issue.
There is of course controversy even with this, as the government is saying unvaccinated people will have to pay for PCR tests to get this (before it was free) and it appears with the Delta variant there is a much higher risk of breakthrough infections.
Personally I think what the UK is doing and giving out rapid antigen tests for free and encouraging people - even those fully vaccinated - to do them often (twice weekly) is a much better way of approaching this, but of course it's expensive and you need to trust people to actually do it.