You could trust them by letting them show you a blood test that showed antigens.
All healthcare resources were stretched thin during the height of the pandemic, your proposal is to add yet another resource-intensive test? For what exactly? So people could skip taking the jab? Don't you see any issues with the cost-benefit analysis of this? Not even accounting for the fact of how easy would be to defraud it.
People were faking vaccination cards, faking a blood test showing antigens would be another very common fraud. You cannot trust people when the consequences are much greater than any individual issue...
> Explanations exist; they have existed for all time; there is always a well-known solution to every human problem — neat, plausible, and wrong.