This is a great question! I decided to write an answer to test my own understanding of the (rather convoluted) forces and interests at play. Assuming my answer makes sense, I hope it will help further the debate on this new proposed taxation rule.
Why can't universities work around the proposed reclassification of tuition remission as taxable income, by not charging tuition?
We don't know that they can't -- I guess we will find out if this proposed new rule becomes law. But with that said, I think working around the rule would require a lot more than just relabeling "tuition" as something else or some similar cosmetic change. The reason is that there is something pretty honest and logical about the current arrangement, which makes me think that changing it would lead to an illogical, dishonest situation, which would be problematic from multiple points of view and create all kinds of undesirable distortions.
My understanding of the current system is as follows. There are four parties involved in the exchange of funds and services when a student goes to graduate school:
A funding entity (e.g. a public or private funding agency or an internal university grant).
The government, acting on behalf of the interests of the rest of society (yeah yeah, I know how naive that sounds).
Each of those parties has its own interests and has something to gain from the transaction, and provides some benefits to the other parties in return:
The university provides the student with an education and professional training, a costly good to provide (requiring physical infrastructure and lots of human capital). In return, it gets the tuition money (from the funding entity, or in some cases from the student him/herself) and the benefit of the research and other intellectual output of the student.
The student gets an education and valuable professional training that is likely to translate to a happier, more fulfilled life with improved employment opportunities. In return, he/she does research and other intellectually taxing work that benefits the university and the rest of society.
The funding entity provides funding to the university to offset (part or all of) its costs of educating the student. In return, it gets nothing, except the knowledge that it has fulfilled its mission of facilitating this process. (After all, funding agencies are set up precisely to be the entities in charge of giving away money to fund research.) This knowledge is used by it to justify its continued existence to the higher-level entities that are the source of its funds.
The government and society get the benefit of the student's research and other intellectual output, and the multiple benefits associated with a more educated populace. In return it gives a tax exemption to the student from paying taxes on the tuition money, which is after all a material benefit provided to the student by the funding entity (except for those rare students paying their own tuitions). This tax exemption is in effect a subsidy or incentive for people attending graduate schools. Of course, as readers of this site are well aware, going to grad school in the US is quite an expensive proposition (in both real costs and opportunity costs) already, to an extent that even deters some people -- probably a lot of people -- from pursuing a graduate degree. Going to grad school still makes economic sense for lots of people, but certainly not for everyone, and requires a lot of sacrifice and patience even from people who do end up benefiting.
Now imagine what happens if the tax exemption in item 4 is eliminated through a unilateral decision by the government, which is only one of the four parties. This will result in a net flow of huge sums of money back to the government. The whole set of incentives of the other parties will be drastically impacted. It is very hard to predict the exact consequences, but likely there will be an equally huge drop in the number of people wanting to go to grad school. Universities will have to go begging for more money from the funding agencies to offset that and restore equilibrium to the system, but the funding agencies don't have more money to give, or they would be giving it already. So: a big, big problem for everybody.
Now, you are proposing that universities simply eliminate tuition to forestall this flow of money back to the government. But then how will they offset the high cost of the education and training they are giving to the students? Charge money from the funding agencies, you say, but just don't call it tuition. Well, the funding agencies have to agree to this. How would the money be charged? By what logical name would you call this transfer of funds (it is after all a subsidy by an external entity of the student's education, which is a benefit to the student and therefore taxable as income under the new rules, at least if it is called "tuition"); and how will you prevent the government's taxation agencies from coming and saying -- quite logically in my opinion -- that this is effectively a form of tuition subsidy whether you call it that or something else, and that the applicable tuition taxes are therefore due? These are all tricky questions. As I said, I don't know if the universities will be able to find a work-around, but it is far from obvious to me that another equally viable arrangement can be found under the proposed new taxation regime.
Finally, I should add that the current arrangement works well not just because each of the four parties gets something in return for what they give, but because each of them gets back much more than what they give. There is a very large net gain of utility to society from the existence of higher education (and graduate schools in particular). That is why it is a resource that is subsidized, in one form or another, in all advanced countries.