It’s their fault in that this could be a $400K expense or a $100K expense and they need to learn the hows and whys. That they didn’t know - now you’ll tell them and hopefully they’ll start to learn quickly.
If they think you’ll spend $10K a year, then no reason to apply to Stony Brook - the room and board alone will cost more.
You can do for sake of doing….but…if that’s truly the budget, it’s not an option.
You’ll have to apply to meets need schools that work for your budget (you’ll have to run the NPCs on many) or more likely, whichever public you can commute to - or a local community college. Many successful kids do this btw - both the Junior College or local college route. There’s many good ones in NJ - Ramapo, Kean, Montclair, Stockton, Rowan, etc.