If you compensate the developers' losses in revenue with tax breaks, how does it shrink the supply?

If it doesn't encourage developers to build more houses, then it does nothing.

We care more about the number of people housed, not merely just people who are housed who can afford their bills.

The original question was "How do you get teachers, garbage collectors, plumbers, nurses, all sorts of manual workers ?" Those people all have some amount of money they can afford to pay their bills. Since a city needs some percentage of those workers, having housing they can afford is a big step towards attracting them.

If you want affordable housing, then build houses.

But where is the line? Rampant development can completely alter the feel and culture of a neighborhood and city. It can take away from those things the existing residents enjoy. Sure more supply can accommodate more humans but they won’t necessarily be the same humans if current residents leave, and it won’t necessarily be the same place afterwards. Desirability (demand) creates scarcity but scarcity can also be desirable in itself for some things. At some threshold, the answer isn’t build more but locate people elsewhere and build a more distributed economy rather than a few concentrated powerful cities or states.

The local government version of “f*** off, we’re full”