The well endowed schools presumably do not care if an individual admit is needy or not, as long as the entire admission class matches up approximately to the target level of financial need. Being well endowed (to the point that even list price students are being subsidized by tens of thousands of dollars per year), they have enough of a margin of error that a few more needy admits than planned will not matter in the big picture. The tipping of the admissions process and criteria to favor the wealthy occurs in the design of the process and criteria, not in the actual admissions readings where the “need blind” claim applies.
The less well endowed schools that promise good financial aid are more likely to have a strict financial aid budget to stay under, so they may have to be explicitly need-aware for at least the marginal admits to avoid going even a few needy students over budget.