The important thing here is that people leaving a city due to unavailability of housing thus stopping the growth in the price of housing is something that we only just see happen, it's a lot less delocalized and it's a long term process, so I really don't see how the real world effect of zero from increased supply is supposed to counteract this second order effect.
The second issue is that while in theory neoliberal YIMBYs are often for public housing, when their plans are put into practice this part is completely ignored. If you see a crucial aspect of your plan being ignored, which is also by far the hardest to implement, and yet don't put it front and center to stop this, it's clear that you're okay for trading that off. Which means that the real world effect is for rent to just keep on increasing.
Whereas cities with progressive solutions (which also include increasing supply, but fittingly with the weakness of the effect don't make it completely central, often seen much more than 0-2% temporary decreases.