The problem with the EU Covid passport is that you need to be inside the EU to get it. Where I am, I could get the Chinese vaccine, but it's not recognized in the EU, so each time I'm travelling to the EU, I must pay to get a test to cross the border. I know you can get vaccinated for free when you're inside, but you must stay there a whole month to get it (2 shots + delay to get the passport).
Note that I've been 4 times to Hungary, and nobody asked me anything at the border. In Serbia, a guy asked︀ me a test, but I was too slow getting it out of my pocket, so︁ he told me to move away. Austria is the worst. I was asked to get︂ out of the car to show my test. MNE don't ask anything, that's cool.