This is not entirely accurate. Even colleges that meet full need sometimes provide packages that families find unaffordable; that is a reason why it is important to check the net price calculator before applying. For my daughter, full need colleges ranged more than 10K from lowest to highest in what they thought we could afford. If the package came out to be more than the student expected, they can get out of the agreement.
HOWEVER, I would be shocked if NYU would be willing to just pause on the acceptance and allow the student to compare offers. Once they say they can’t afford it, they doubt that they can come back on March 31st and say “well NYU, you turned out to be the best offer I got so I am coming after all.”