The universities I know of that have free-tuition programs do phase them out gradually, to try to avoid too much weirdness that would come from a hard threshold

The universities I know of that have free-tuition programs do phase them out gradually, to try to avoid too much weirdness that would come from a hard threshold. For example, Harvard charges no tuition to students from households making under $65,000, but above $65,000 tuition phases in slowly, reaching full tuition only at $150,000+. However I do agree your proposed solution is better. Have universalist public programs that everyone can take advantage of regardless of income (tuition-free public university, no-charge public healthcare, etc.), and then instead of mucking with means-tested benefits and aid phase-outs as a way of indirectly taxing the rich, just tax them through actual taxes, the old-fashioned way.