When it was me, I really wanted to leave home and go to a fancy dream school (Georgetown in my case). But unlike your kid, I actually HAD the test scores and grades to get into a school like that.
But the money wasn’t there. So I didn’t even apply to any country club schools like Gtown.
So I lived at home and commuted to a non-fancy local college I got a scholarship at. I also started working PT and summer jobs in 8th grade and continued that through college graduation. It was what it was, and it worked out fine for me.
You are offering your kid a full ride scholarship to live away at a place like Purdue, which would have been an incomprehensible luxury to me. And in technical fields like CS, frankly a big state school is just as good or better than many fancier privates. For CS, USNWR says it IS better (#20) than NW (#30). Put the dollars aside. TBH, your kid simply is not going to get into a school with a better rated CS program – Purdue might actually be a little reach-ey for CS.
Tell your kid he is lucky, should appreciate his blessings, should shut the front door, and grow up.
The deal was you’d pay for one (and only one) fancy full priced private – NW. He didn’t get in. Too bad. Maybe he should have worked to get better grades and test scores (like I did for Gtown) if he really wanted to take advantage of that opportunity and generous offer. But he didn’t, so that ship has sailed. Choices, consequences.
His budget is a full ride to an in-state school. That is WAY more than what most kids get. Tell him he can spend that at whatever school he (i) can get into and (ii) can make work with that budget. End of story.