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things which are the proper domain of the public sector
The division in politics is over what that set consists of. For my money, Hayek is a bit too much of a socialist :) In any case, it is not necessary for private sector to mean the extraction of profit. If a not for profit health insurer (or school, or library, or bicycle factory or any other of myriad things the communists think are too important to be left to the choice of citizens) does a better job than a for profit one, the former will put the latter out of business.
Hmmm, I'd rather keep profits out of the providing of things which are the proper domain of the public sector.