I think your scenario above is a non-starter. MIT vs UNT? Duh, unless they want to play jazz trumpet that choice is obvious. I think the more realistic question would be UNT vs TxSt or UH or even Tech. Not a lot of difference in Tech and UNT, but tons of kids go there because it has that 'big time' label with Big12 sports. Even your MIT vs Alabama is silly. Anyone who knows anything about academics knows that Alabama most SEC schools in generally are not very strong academic schools. (Now most of the Big10, that is a totally different story).
I think the effect comes in when kids and parents are considering similar schools. Even if a kid isn't into football or basketball, they may have lots of friends that are and get excited about a school because of the buzz.
I remember what George Mason experienced after its tournament run a few years ago. It was pretty amazing. I would almost 100% guarantee they didn't steal any kids away from MIT or Stanford, but both the number and quality of their applications went up dramatically. If UNT were to capture that type of attention, we wouldn't be keeping your kid in state if she had a MIT scholarship in hand, but some kids that might have decided on OKSt, Tech, or Colorado might decide to stay home.