We need to separate the question whether it may be done constitutionally, in principle, on the one hand, and the question whether a government has the requisite competence to do it on the other.
Federal governments have been pretty mediocre for quite a long time. And they'll happily remain that way whilever most of the public accept (or give up in frustration while watching) the blame-shifting games. Just look at quarantine.
But be in no doubt: A pandemic is the paradigm example of a matter that falls squarely within the nationhood power and, even if it didn't, the Commonwealth has more than enough power to 'encourage' the states to implement what the Commonwealth wants (e.g. some semblance of 'order' in the aged care sector). It's a political choice and - I'll say it - political cowardice not to.