What's the point of taxing tuition waivers?

This post is about graduate students, the tax reform bill and Congressional politics - don't want to read about it? No worries, I won't be offended. 

Last week, the House leadership released the Tax Cuts and Jobs Act, which is their approach to tax reform. There are lots of various constituencies that are irritated by this proposal (e.g. those who are adopting children will no longer have access tax relief), but it's a surprise when there's something that affects graduate students. From Inside Higher Ed's Andrew Kreighbaum, a short summary of the bill's effects on students (emphasis mine): 

Students -- especially older, part-time and graduate students -- would also see negative consequences from provisions of the Republican proposal, higher ed advocates said. The bill restructures the American Opportunity Tax Credit, eliminating tax benefits for students who take more than five years to graduate, as well as part-time and graduate students. And it repeals the Lifetime Learning Credit, which is used by grad students, workers who need retraining and part-time students and nontraditional undergrads who take more than four years to graduate. 

The proposal would also eliminate a provision of the tax code used by many universities to waive the cost of tuition for graduate students filling positions like teaching assistantships. If the proposal were to go through, those institutions wouldn't be able to waive tuition costs without imposing new taxable income on grad students, said Steven Bloom, director of government relations at the American Council on Education.

I find this proposal rather bizarre - how much could the government stand to make from this? I have always found this tuition waiver to be somewhat of an accounting fiction, i.e. a department of chemistry at a university enrolls a graduate student, and the university will charge the student tuition, and issue a tuition waiver? And so I presume that there is money that is transferred from column A to column B in the university's books, but it's not as if the student sees any amount of that money.

Let's also stipulate this: if this were to proceed the way that it looks to proceed, graduate students would be devastated, as their taxable income would go from $20-30k to (depending on the size of the tuition waiver) $70-80k. According to the tax tables, that's a move from somewhere in the $2000 range to the $8000 range, depending on exemptions, deductions and the like. That would be brutal for anyone, and people would be forced out of graduate school quickly.

But here's my real question: why is the House GOP doing this? Is there really that much money to be gained from this? Is it (as lots of people on Twitter saying), punishing one's political enemies? And if this were to actually be passed into law (something relatively unlikely, I suspect), wouldn't lowering charged graduate school tuition to zero be a way to get around this? especially if the money doesn't actually exist?

Readers, tell me where I am wrong.