The devil is in the details and the situation. Let me guess, the 3 Ivys did not include H, Y, or P.
10 years ago, my son went through the same drill thing with Johns Hopkins, Amherst and Williams. He was told he was good to go with Cornell, and Tufts which we were fine with as he was an engineering major and they both had outstanding engineering programs. Cornell and Tufts were recruiting him hard. Harvard was offering but they really wanted him to retake the SAT too improve his score and the team AI. He retook the test as a way to expand his choices because at that time he wasn't sure. A funny thing happened after that. A couple days later after he retook the test, he had made his college decision based on the engineering programs rather than the school and their required SAT scores. He did improve his 2nd score 150 points exactly, but it didn't matter. He made up his mind.
Bottom line is you can expect some of these schools to come back to you wanting more points to help the recruit and team AI. It also helps the coaches position with the Admissions Committee if he presents a candidate that has gone out of his way to improve his position. He can best help you if you've helped yourself.
Just my experience....
His reads were Yale, Dartmouth & Brown. Harvard spoke with him but never asked for scores.