Our “problem” is that we can produce a sufficient quantity of goods and services without employing the services of all willing workers.That is the 'paradox of productivity' and is a central question to policy. Because if there is intrinsically insufficient demand in an economy then the private sector can never reach maximum output on its own. Even with a simple model the equilibrium falls short of maximum output.
If we do have a 'paradox of productivity' then there must be an active government sector keeping demand up so that the private sector will perform at maximum output. The public sector has to at least pay for some jobs. There is no choice in the matter.