“That doesn’t make sense if the reason for the poverty level isn’t related to education.”
It is not so difficult. If the State can show that students of a Yeshiva are not self-sufficient and rely on social programs, the Yeshiva will not be able to make a strong case based on Yoder. The State does not need to show why they are impoverished, just that a disproportionate are. The burden of arguing that they should not be regulated will fall on the Yeshiva as a result.
FYI, if a school were to argue that they teach their students to avoid gainful employment and instead to rely on social welfare, the State may decide they have the right to shut-down such a school. So if in theory you were to argue that we teach a belief that the students should go learn, which will result in being funded via social programs, nevermind the chillul hashem such an argument would cause, it would be tossed out of court so fast.