Any government can create such database without the help of a blockchain technology.
Blockchain will just make the database a little bit more secure and transparent,I don't see any other benefits of such implementation.
By the way,tests aren't 100% accurate and there are doubts about this "coronavirus immunity" theory.
There were many people that were cured from the virus,but got infected again.
Creating a totalitarian society ruled by technologies,just to fight a pandemic sounds really dangerous to me.