If we stand for the "right" of Nazis to compel private actors to do business with them (and holy shit does that violate the typical libertarian-capitalist positions of the American Right!), why don't we stand for the actual right of Communists, revolutionary socialists, democratic socialists, and anarchists to operate openly at all?
AFAICT, while the near total lack of government enforcement (and lack of appeals in the few relevant cases that have gone to court) have prevented many or high-level court decisions on the overall constitutionality of the Communist Control Act, the only federal court to rule on that issue ruled it unconstitutional, and the only Supreme Court decision relating to the Act held that the Act did not prevent the Communist Party from participating in the New York State unemployment scheme.
So, it's not clear to me that the Communist Control Act has any substantive legal force, and it's pretty clear that whatever it seems to say on its face, the force it does have is not actually outlawing the Communist Party.
Communist Control Act, 50 U.S. Code ยง 841
>The Congress finds and declares that the Communist Party of the United States, although purportedly a political party, is in fact an instrumentality of a conspiracy to overthrow the Government of the United States. It constitutes an authoritarian dictatorship within a republic, demanding for itself the rights and privileges accorded to political parties, but denying to all others the liberties guaranteed by the Constitution.
https://www.law.cornell.edu/uscode/text/50/841