Market Rents are not a benchmark of zero rent-cost gap - only a complete idiot could make such a conclusion.
"why does the private sector need to charge twice as much?":
Because private rent makes 100% profit
Because private rent is less efficient with higher per unit costs
Because private rent has to support a mortgage rather than a lower cost loan
Because the individual can not be as efficient as the collective in providing for mass provision (e.g. could Henry Ford make all the cars by himself - no he needed the input from thousands of others to make it stack up)
Social housing is cheaper because it is more efficient and more effective in terms of providing low-cost housing.
Private housing is more expensive for the opposing reasons.