One major issue with counting the number of people who are homeless is that most of them are not on the street, but are instead relying on friends and family for shelter. The people who live on the street are the spear tip of a much larger and broader group of people who are facing various degrees of crisis in finding housing. It is difficult to come up with a method that captures data about all the people who, for example, were evicted and spent several months living on couches and in their car.

Right, this is because the homelessness industry needs to justify its existence with ever inflated numbers. Someone down on their luck and staying with parents or siblings is not homeless and does not need intervention for shelter. In general, we should encourage extended family living as its economically easy and socially good. These 'homeless' need economic help, not shelter, and its likely that, due to their proximity to family or friends, this help can be provided in large part by the private sector as long as government policy encouraged the creation of jobs. Really, when civilians say homeless they mean what researchers call unsheltered, the subset of homeless living on the street. This group has much higher needs, typically around mental health and drugs

Why is that a major issue? The regular Census can capture information like "did you provide transient housing in the past year" , and "if you hae had a transient resident living with you, please contact XXXXX so we can directly include that person in the Census"

There's a small portion of chronically homeless that don't want to be in any systems though; whether it's because they are paranoid, have a criminal history, have a debt or potential of, etc.