Private and public are both viable models of operations with some applicability overlap. Private doesn’t necessarily pursue profit first, despite US literally enforcing it.
Basic needs that are either unchanging or change very slowly are the purview of public policy. Healthcare, infrastructure, etc. Privatize it and you’ll have a catastrophe.
Basic needs that benefit from variation and supply elasticity with a necessary baseline is where hybrid model works well. Public entrepreneurship provides variation, regulations or public enterprises cover baseline. Agriculture is a great example of such overlap. Private-only agriculture leads to profiteering on basic human need. Public-only agriculture leads to famines due to incompetence, malice, or lack of elasticity.
Desires that people can live without and can change on a whim is where private innovation thrives. Be it a product to sell or a charity project to pursue. Some of the results of said innovation can and will become matters of public interest. Forbid private enterprise here, and you’ll end up in a bleak reality of North Korea.
We literally had a case of “public everything” half a century ago and it didn’t fucking work. It needed serfdom and insane amounts of natural resources to prop itself up. It also left a mafia-led capitalism in its wake.
We also have a live case of blind trust in markets, as if information was immediately available everywhere. It leads to a very similar looking outcome.