If you're not looking for a candidate for an entry level opening, then I think the school plays a smaller role, but it could still be the foot in the door that you need to get the interview.
Hundreds of applicants could just be a bigger company- not sure why that makes the view short sighted? They have to be filtered somehow. You can't expect everyone to be given a shot. If you know a school has a more challenging/higher rated program, why wouldn't you give preference to that applicant? Again, referring mostly to entry level here. My company added a huge customer at the beginning of the year and there were openings for 60-70 managers. A couple of people do the intial screens and phone interviews for this area of the country- I doubt all applicants got a call back. Perfect example of when the well known school may help you get the interview.
I'm not advocating going into huge debt to get a better name on your degree. There are multiple ways around footing the full bill for a big name school. Again, have to look at the cost versus benefit by asking about average salaries, etc, but I wouldn't rule out the option based on cost alone.