Does a child's business income go on the parents' return?

Usually no for earned business income. A dependent child's earned income is generally the child's income, even if the parents still claim the child as a dependent.

Original example:

Theo, age 12, earns net profit from weekend lawn cleanup work. His parents provide his housing, food, and school costs. Theo may still be their dependent, but the lawn profit belongs in Theo's own filing analysis.

If a dependent is required to file and cannot file personally, a parent or guardian may need to file for the dependent. That does not turn the income into the parents' income.