So who comes and murders the hot dog vendor to death for operating a stand in his private park without giving up his cut? Where does this homeless person sleep while he's raising his capital and picking himself up by his bootstraps. Your entire argument is predicated on the existence of land that is free to use by others.
It doesn't take long for them to get enough money to build a hot dog stand.
What do you base this on? If everything is privately owned and they have nothing to offer but labor, which is devalued or valueless in our hypothetical robot-run world, where do they get this money?
If he's starting with nothing, he's only got a few weeks to build up the capital to start his hotdog stand, while diverting some of the money to food, water, and rent (there is no public land, remember). If he fails to raise the money or misjudges his market, he starves to death, right?
I assume that in a world where labor had little to no value, you'd never be that homeless man, right?
[To keep this discussion on track, I'm not some authoritarian statist or communist or anything. I'm only pointing out that your solution to this thought experiment isn't very well thought out.]