yes it does
if the injections works at intended (protect you, fully, like any real vaccine) there is no need for a pass as your are protected and risk free
if the injection doesn't work, same, no need for a pass.
aka if you are injected and still transmitting aka not blocking the spread, the pass is useless but to identify you once the side effects really kick in (according to estimate 3 years after injection in humans, personnally waiting for at least 7 generations to be safe and secure).
No. Your assertion is only valid if everyone has been vaccinated. In the real world, some people have been vaccinated and some haven't. If the vaccine is effective, then the pass can be used to prove that someone is 'safe'. Without the pass, there is no proof.