There already is something kind of like that: It's called the Common Application. You fill out a single application and choose which schools you want to receive it. A little under 1,000 schools use it, so if one of the school you're interested in accepts it then you can easily click a few boxes to send it to a bunch more with no extra work.

As for paying the application fee, it's not too hard to get a waiver. All you have to do is get your guidance counselor to sign off on a waiver form. (That is actually not just for the Common App either: nearly every school will accept that sort of affirmation of need and waive the fee)

The primary reason fees exist is to make them high enough that students who aren't all that interested in attending don't bother to apply. Believe it or not, for most schools the application fee really doesn't impact their budget much. Take a mid-size school with 20,000 students and a freshman class of 3,000. They might get 12,000 applications that progress far enough that the student submits a fee. Take away 3,000 for fee waivers, and average a $60 fee, and it adds up to $540,000. Seems like a fair bit of money, Given a staff 20, marketing budget, etc., it's a very minor consideration.

Sure, it's not nothing, but having lots of not-very-interested applicants makes it extremely hard to make a target enrollment goal because you end up accepting so many students that won't attend. So you either come under your goal because you accepted uninterested students, or you accept too many to make up for it, making it look like your less competitive at the same time you risk enrolling more students that the school can handle on such short notice (I've seen that happen, it's not good for the school or the students).

Source: I work in higher ed analytics.