Biden and Obama are called socialists because Americans were trained to fear communism and socialism during the Cold War. There is nothing about their political stance that is socialist. Conservatives have just learned that they can score easy points by calling anyone left of Ronald Reagan a marxist
If it is redistributing wealth, it's socialist. Taking taxpayer money from person A and paying it out as a subsidy (even if laundered through a private company) to benefit person B is socialism.
Anytime you are redistributing wealth under the threat of the government's right to licit first use of force, you're engaging in socialism.
Socialism as defined by socialists is social ownership of production.
What you're describing is socialism as defined by its libertarian intellectual opponents, and that's a pretty broad definition (any taxation followed by spending is socialism? That's pretty odd... so all military and police forces are socialist?)
So, fine, if that's what you choose to believe, but it's certainly not a policy proposal that is advocated by actual socialists. Even the soft left wing of the Democratic party found/finds the ACA objectionable.
In fact it's explicitly the kind of policy that liberals advocate in order to avoid socialist policy measures and to avoid being called socialists.
“ Socialism is a political, social, and economic philosophy encompassing a range of economic and social systems characterised by social ownership of the means of production.”
https://en.m.wikipedia.org/wiki/Socialism
You should read the rest of that wikipedia article instead of just the first line.
There are many variations discussed in the article. What you should look for are the market varieties of socialism, especially the ones where the objective is to achieve socialism by following peaceful mechanisms of achieving socialistic goals instead of achieving socialism via revolution.
