Add me to the list of counterexamples to your claim. While my family was more lower-middle-class than poor, I was (evidently, according to the financial aid office) the poorest person in my graduating class. And my SAT score was likely the reason for that. (I increased my score by 200 points to a perfect score using practice books.) I was also admitted to Harvard but didn’t attend because of the insufficient (at the time) financial aid that was offered.