What model is a suitable model for zero-constrained variables?

After reading the answer in How do I normalize a bimodal distribution?, I began to think about the concept more broadly. With OLS on zero-constrained variables, the residuals signal bias, as the zero-conditional mean fails to hold (if I’m not mistaken).

Question

Aside from demeaning, what is the conventional approach for zero-constrained variables, and what (if anything) changes if the zero-constrained variable is a response / explanatory variable?