Demonstrably false. The commissioner's order (which I linked above) never required them to be fired. The order solely required they be kept off the premise which is not the same as fired. Sure somebody could fire them and then claim the health order told them to but that wouldn't make it truthful; the health code is very clear on violations being a fines / misdemeanors.

> But they could still go to grocery stores and baseball games, because those places aren’t important.

Sure. Perhaps the agency has the power to stop people from doing that. It's not explicitly listed though.

> An agency, no matter how well–meaning they are, is not allowed to take any action not on their list.

The legislature has delegated a vague do everything to the agency. So, everything is on their list ...

"Except as otherwise provided by law, the department shall have jurisdiction to regulate all matters affecting health in the city of New York and to perform all those functions and operations performed by the city that relate to the health of the people of the city, including but not limited to the mental health, intellectual and developmental disability, alcoholism and substance abuse-related needs of the people of the city. The jurisdiction of the department shall include but not be limited to the following:"

However, the Health Department does have explicit control over "(5) ... operation of facilities by other agencies of the city;". Which is exactly what they were doing. They were telling other agencies that they can't operate their facilities with unvaccinated individuals. They also tailored the order to be as narrow as it needed to be. The unvaccinated individuals are still allowed to work, just not on premise which is how a disease would get transmitted at those facilities.

[1]: https://www1.nyc.gov/assets/doh/downloads/pdf/covid/covid-19...