I don't think it's as obvious or causal as you imply. If every wage increase resulted in a corresponding increase in rent to force people back into poverty it would never make sense to increase minimum wage, or even wages at all.

But that's not the case is it? Wages stagnate, increase or decrease all the time without changes in rent. Rent goes up without changes in wages all the time.

Sure, a local across the board increase in wages might drive up demand and might cause some portion of rent increase, but it's also just as likely to drive up housing prices as people get out of the rental market.

Throwing up your hands and saying "we can't pay them more because landlords will take it all" is disingenuous.