There already exists the International Certificate of Vaccination or Prophylaxis. It is internationally recognised, and some countries already require to see one for certain diseases (e.g., yellow fever) before they will permit entry. I really don't see the problem with using that for travel.
As for pubs that require one to use an app... "other pubs are available". I don't take my mobile to the pub - partly because if I do, I just know that work will "only want a minute of my time", and partly so no b***er can nick it.