While I agree with your reasoning, I think the generally accepted understanding of "working class" is of skilled and unskilled laborers that are distinct from the "professional class" that tend to work in offices and don't work so much with their hands. A plumber running a small business employing other plumbers with a massive cash assets could be a lot farther from being homeless than a senior lawyer with multiple houses and expensive cars but the plumber will always be perceived as "working class" in contrast to the lawyer's "professional class".