I appreciate the sentiment, but the public sector supporting the private is not "socialism." Socialism describes an economic formation where public ownership is primary in an economy, ie where large firms are publicly owned and controlled. Segments of an economy cannot be Socialist or Capitalist just like an arm cannot be a human, it can only exist in the context of the whole.

Socialism, in reality, refers to a broader economy where public ownership is primary, while Capitalism refers to a broader economy where private ownership is primary. All Socialist societies have had public and private Capital, and all Capitalist societies have had public and private Capital, it matters most which one has the power.

I recommend reading my post here on common problems people run into when determining Modes of Production.

That's state welfare/insurance, not socialism.