You don't understand. Almost nobody actually thinks about this in the right way, but it's actually basic economics.
Salt.
We used to fight wars for salt, but now it's literally given away for free (in restaurants).
If "robots produce so much value" then all that value will have approximately zero marginal cost. You won't need to distribute profits, you can simply distribute food and stuff and housing because they're cheap enough to be essentially free.
But obviously, not everything will be free. Just look at the famous "cost of education vs TV chart" [1]. Things that are mostly expensive: regulated (education, medicine, law) and positional (housing / land - everyone wants to live in good places!) goods. Things that are mostly cheap: things that are mass produced in factories (economies of scale, automation). Robots might move food & clothing into the "cheap" category but otherwise won't really move the needle, unless we radically rethink regulation.
[1] https://kottke.org/19/02/cheap-tvs-and-exorbitant-education-...