Right. It provided proof that the authorities were willing to lie “for the greater good”. So at that point their statements cannot be seen as truthful but only “possibly good for us to believe” (they think).
And as soon as people accepted that possibility, combine that with their personal estimate of how often “greater good” might conflict with what they personally would consider the actual good of the country or themselves, and you have a situation where many people don’t comply with the recommendations.
Lying to the public is a bad long term strategy for achieving cooperation.
But a lot of these people (perhaps justifiably) already viewed media and public officials as perfectly willing to lie to them or ignore their concerns - this extends beyond the pandemic itself and is part of a broader trend of mutual distrust.
At this point I’m not even sure it matters. The states with the lowest vaccination rates are never going to have vaccine passports anyways. Mentally and culturally the pandemic is over for most people, vaccinated or not.