I agree on the land value tax part of what you said, though.
What would incentivize the building of low or medium income housing, when higher profits can be made from high income housing.
In the US and many other countries, the reason is definitely local government restrictions. Developers would love to build more high-rise apartments - it's very profitable to them - but they're not allowed to in many areas. In the area I'm living in, it is illegal to do it.
The incentive picture is not as you're portraying, and we can look to other industries for counter examples. If there's profit to make, the market will fill those shoes, even if there's "more profit" (which I don't agree is the case always due to volume) serving wealthier customers. Consider the market for low-margin cheap restaurants in poor areas, and the market for high-end dining in rich areas. These are both very well served markets. The market will almost always find profits where they're to be had, especially in an industry like real estate development.