I believe the short answer is no. Most universities have merit scholarships (some are guaranteed others are competitive), and financial aid. There are some schools that have a merit component to some of their need based grants or scholarships but they will not be your most competitive universities. One could argue that competitive scholarships fall under this category but there are usually very few of them.
One danger of that type of reasoning is that if you don’t get that super secret scholarship then do they not want you enough? This that time of year where reality can really sink in for a number of very good students. They end up accepted to universities they simply can’t afford to attend. If they haven’t planned for a school that they know they can afford they up either going to a school they really don’t want to attend or taking a gap year and regrouping or the worst option, the family takes on huge debt to allow the student to follow their dream.