I think you're using "wealth" to mean "assets" which I don't think is a useful number for any of these comparisons. In the example you give where a person has a house which declined in value and some student loans, they do have a negative net worth but they are also worse off than the person with 0 assets and 0 debts.
Having the house that they need to make payments on means that not only do they have 0 or negative net worth, they also have strict cash flow requirements which the person with 0 assets and 0 debts does not have.
I mostly agree with you, but I think you're going a bit too far: Someone living on the street, having no income, would have 0 assets and might have 0 debt (until they need medical attention...).
For someone with a family, struggling to keep the house is certainly better than living on the street. Still, being in a credit crunch could of course lead to that family being evicted, so yes, negative net worth isn't great.
Having the house that they need to make payments on means that not only do they have 0 or negative net worth, they also have strict cash flow requirements which the person with 0 assets and 0 debts does not have.