Most of the argument is whether this requirement is in the "employee must wash hands before surgery" category, which is about the patients' right to be safe not the employee's (in theory the patients didn't "choose" to be there).
The people complaining are reacting as if the ruling sounds like "male high school teachers get vasectomies, to keep teenage pregnancies down", because the effects are "permanent, * upto six months" and directly to your body without it being an on-the-job requirement.
The part that makes me leery is that these sort of "are your vaccines upto-date" checklists have always existed, but this time it is controversial (or maybe it was for MMR - but it wasn't news).
this is also the first time where the vaccine industry and the broader public health space have been put under this kind of scrutiny and the combination of politicization, outright lies and deception over masks and policy in both directions and the overall visibility of the sausage making process happens to be occurring alongside a decline of trust in institutional expertise. when most people got their tetanus shots, the faces of the medical profession were not acting as explicitly political agents.
Nearly every common-sense measure implemented during this pandemic has been derided by the right as some sort of unacceptable infringement on individual rights.
Not saying those in power have gotten it all right 100% of the time; I'm especially disappointed with the mask fiasco you mention, but... c'mon. Doctors are not acting as "political agents" in the vast majority of situations that have caused the US's pandemic response to be as lacking as it's been.
It's not common sense if a plurality of people don't agree with it. Denying children in-person schooling for a year wasn't "common sense", cancelling elective procedures like cancer screenings wasn't "common sense", forcing people to wear masks outdoors wasn't "common sense", and so on. Lots of people were denied rights in these cases and would consider those measures unacceptable.
something, something, consequences.
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If expect there are certain Catholic organizations who would fire you if you admitted to having had an abortion.
... Which goes to the court ruling that's the topic of the article: Congress has the power to decide these issues.