With that model, if the government are going to be effectively paying whatever the landlords demand for rent, then the best solution would be for the government

With that model, if the government are going to be effectively paying whatever the landlords demand for rent, then the best solution would be for the government to seize rental housing from the landlords (perhaps with some token compensation so they don't complain too much) and run it as a non-profit rental service, i.e. social housing for all. Obviously not the best solution for the landlords, but, really, so what? Removing a class of parasites from the housing system would be a feature, not a bug.