First of all, it's Vienna, not Venice. And you don't need a counterfactual Vienna to compare it to cities of similar size and economic characteristics.
> the best they could do would be to get rid of the (actually quite small) cut that landlords take for themselves
Again, I think this claim is easily contradicted by the success of Vienna's housing market. The thing is, if you took the market as it is today, and the government just bought all the housing properties at market rates, then yes you would only see savings in terms of the profit margin which is currently enjoyed by landlords. But if the government takes the role of providing housing as a public good, it can change the incentive structure of the market in a way which benefits renters.
For instance, if your goal is to maximize rent profits, you will make decisions to maximize the year-over-year increase in rent, and you will build more units in a way which maximizes the rental price per unit. That's how you end up with so many shoebox-sized luxury condos in Toronto that nobody really wants and an under-served affordable housing market. If you treat housing as a public good, you make decisions to keep rent prices stable, and you build more housing to serve the need rather than the profit motive.