The orthodox theory says that, in a competitive marketplace, producers should not be able to "capture" gains in the wealth of consumers. If prices of a good rise, it's because consumers are purchasing more of the good, and the good has increasing marginal cost to produce. (I ignore some 2nd-order effects related to wages as a production cost).

That's the orthodox theory. You shouldn't buy it wholesale.

On the other hand, it would be a very heterodox environment indeed if producers were able to steal all the increases in wealth from consumers. That could only happen in an environment of no competition, not imperfect competition. Or an environment with 100% occupancy and no ability to build more units of housing.

Also, there would need to be no substitutes for the good. For instance, that people couldn't move between cities.

Surely the game is, to some extent, rigged. But can it be rigged so badly that it is possible to believe that transferring wealth from richer to poorer will not lead to an increase in the purchasing power of poorer people?

I don't have a decided opinion on guaranteed minimum income, but to deny that it would have a marked effect on purchasing power of the recipients of the wealth requires one to posit a truly exceptional economic theory. To attack minimum income on its efficacy of all things: that is either an ill-considered or truly radical view.

If minimum income can't help, what possibly could? The only option left would be the confiscation and redistribution of capital.