I'm in agreement with many of your points, but there is one point that I think you've missed. You wrote that:
"But in a free market, there is no reason why labor should underform capital."
Let's consider the following situation: Mortimer Moneybags has built a factory that builds cars. It cost him billions and billions of dollars, but it is stocked with robots that do the whole shebang. Trucks drive up with raw materials, robots unload the trucks, stuff goes into the factory, and shiny new cars come out of the factory. It needs no workers, except for one guy whose sole job it is to push a big red button that says "STOP" if the computers say that something is wrong. That's his only job, and he never actually pushes the button; he just sits in his chair all day waiting for a notice that never comes.
Let's say that this factory earns a trillion dollars a year for the cars it makes. Should capital and labor share the profits equally? That is, should the worker get half a trillion and Mortimer Moneybags get half a trillion? Obviously not. Obviously capital is responsible for almost all the production of the factory, so the owner of the capital (Mortimer) deserves almost all of the profit.
A few centuries ago, almost all the productivity came directly from labor. Over time, though, more and more productivity is done by capital (i.e., machines). Hence it is only natural that more and more of the profit goes to the owners of the capital.
It won't be long before machines provide the great majority of the goods and services we enjoy, and workers will contribute very little to the creation of those goods and services.
I do not conclude that it is right and proper that the rich get richer. I conclude that we need a tax system that transfers wealth from the rich to the poor.