if one is homeless, it's not really due to inflation, but due to their productivity not keeping up. Globalization has moved manual labour and low skilled labour away from the USA, and those people affected were not given help to re-skill into other work, and most have to move to providing low-skill services (like hospitality, casuals etc), which is notoriously low pay.

Blaming inflation (or how inflation is calculated) for a societal problem is a distraction.

Shelter is in the CPI.