I dunno, man. Let's talk about this.
Supreme Court ruled in 1905 in Jacobson V. Massachusetts that states' health departments absolutely have the right to vax your ass. However, the majority opinion issued basically says you can't do this through OSHA:
“Although Congress has indisputably given OSHA the power to regulate occupational dangers, it has not given that agency the power to regulate public health more broadly,” the court wrote in an unsigned opinion.
Which, if you look at the OSH Act, does not cover "workplace hazards regulated by another federal agency". So from a legal precedent standpoint it's pretty clear: if you want vaccines, you gotta come at it through the health department, not OSHA. Which means you gotta come at it through HHS, which means "people who breathe" not "people who work for a large employer." This is why "people who work indirectly for the HHS" was upheld.
Letting OSHA decide what vaccines you get would be problematic. It gives them the legal clearance to mandate any hazard, not just work-related hazards. It gives them the legal clearance to address your habits outside of work, not just at work.
The Biden administration went at this knowing it would be shut down, knowing the libruls would lose their minds, and knowing they lacked the juice to mandate vaccines for the whole goddamn country at every level. To me, this is trial lawyers screaming about McDonald's hot coffee.