To be honest though, I think this is a net positive for conservatism and federalism, and a net negative for progressivism in the long run.
Societal ideas begin and, at first, seem promising only for them to grow and grow until, after a century, die from their own crushing weight. This is exactly where the idea of large centrally control federal government progressivism is right now. It is starting to die from its bloated weight and we are organically moving back towards federalism.
Beliefs become dogma once challenged, and this mandate is one hell of a challenge. If you look at the news, this mandate (overnight) has, like I predicted, only strengthened the resolve of those even mildly against it. Biden is about to get mired down in the courts over this, but the real question is, if he prevails, how will he enforce it?
He does not have the resources at the federal level to actually enforce this, just like Trump did not have the resources to deal with illegal aliens effectively, without state cooperation. So Biden needs to rely on state health inspectors to enforce this. But, just like so many blue states passed sanctuary immigration policies, refusing to allow their resources to help ICE, so many red states will just do the same here. In all of the red states, this will be a mandate without any teeth and become a joke.
This will just produce another organic separation of states away from the federal government, a trend, by the way, started by blue states with their sanctuary laws. If presidents continue to do things like this, eventually enough people on the right and the left will just say, "I may agree with the current guy, but the next I probably wont and would rather not give him the power to interfere with my state even if that power benefits me now."
If you disagree, just look at all of the sanctuary measures already in place against helping with federal enforcement. We have sanctuary immigration policies, sanctuary gun enforcement policies, sanctuary marijuana (and other lite drug) policies, and now, soon to be, sanctuary health enforcement policies. We are slowly moving back towards federalism, and the tighter the federal government's grip on centralized power (regardless of party), the more sand falls through their fingers.