The president need not take full responsibility for the regulations. He can always say that he’s just following the Supreme Court’s ruling, and that, if anything, he has tried to make the rules less onerous.
Members of Congress can say there’s nothing they can do. Even if they oppose the regulations, as almost all Republicans and many Democrats do, they can’t stop them unless they amass a veto-proof supermajority in both houses.
The justices can say that they’re just interpreting the laws on the books, and have given the executive branch plenty of flexibility to make the rules less burdensome.
If you lose your job because of these regulations, how will you know who to blame, even if you follow politics closely? Which bums will you try to throw out of office? How will you go about trying to change things? We could well end up with a far-reaching, slightly bonkers policy subject to no real democratic review.