The rationale in that case doesn't apply here: the COVID vaccines don't stop the spread of the virus. Because COVID vaccines can't actually stop the spread of the disease under any plausible level of vaccination (the R0 of the virus is too high), vaccines can serve only to safeguard the health of individuals, and we generally don't use heavy-handed nation-wide mandates to force people to do things that are merely good for them health-wise on an individual basis.

The same reasoning that justifies this mandate would justify a state-imposed limit on daily calorie consumption, a state-imposed exercise requirement, and all sorts of other coercive measures that, while good for people's health, are flagrant violations of principles of bodily autonomy and informed consent for medical treatment.

This isn't about the vaccine and it's not even about COVID. This is about one political tribe using the state to hurt the other political tribe.