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The 10th amendment doesn’t change the supremacy clause. It simply makes explicit what’s implicit in the supremacy clause: federal law takes precedence over any and all state laws and constitutions when they are made in pursuance of the US Constitution, so the 10th amendment clarifies that if it’s not a power granted to the federal government by the US Constitution, then it’s reserved for the states. To invoke the 10th amendment in this case you would have to prove the federal government is acting beyond its constitutional scope, which would require either proving it’s going beyond EMTALA or that EMTALA itself is unconstitutional. They are not making either claim in this case.
Viruses evolve, some quite quickly. The flu isn’t the fastest (looking at you, HIV), but it’s up there. Over time, existing vaccines train your body to fight something that doesn’t quite match what’s in the wild (i.e. efficacy goes down with time). That’s why there’s a different seasonal flu vaccine every year.
They create flu vaccines on a yearly cycle, and a pandemic can kick off in a matter of weeks and months, so if it doesn’t match the preplanned cycle, they’ll have to invest more resources to creating the most up to date vaccine off-cycle.