The origin of public interest theory can be traced to Pigou. I think land use control is based on that theory. However, economists appear to be more interested in public choice theories in recent decades. That also has implications to land use control and might be the key to understanding corruption in land use control.
There are issues that should be addressed through land use controls. These issues have their basis in technological external diseconomies (and economies). Specifically, we are addressing pollution (such as air and water quality differentials), noise, and congestion. Assertions about public "interest" that are found without reference to these diseconomies should not be considered as serious.
As Feng Deng has explained, there is a line of knowledge about public interest deriving from economics. Summarised rather loosely, this holds that the public interest is served by well-functioning competitive markets, so long as externalities, such as those identified by Peter Colwell, are accommodated.
There is a separate line of scholarship deriving from public administration studies. I summarised this work in my own doctoral dissertation. I argued that there are different meanings of the term. I argued that the term is always contextual (depends upon culture and circumstances of the particular polity being studied) but is never meaningless and that there are solid foundations on which a government can formalise an understanding of the public interest at a point in time for a particular purpose.