This argument, often repeated by people opposed to the unionization of grad students, is a non sequitur. It jumps from the professor/student (i.e. mentor/mentee) relationship directly to the supervisor/employee relationship without justifying a connection between the two. It's a false equivalence, since the relationship between a professor and her/his student is primarily academic, while the relationship between the university as an institution and the student as a worker is primarily administrative. The latter relationship is what the NLRB decision addresses.
The (fear-mongering) claim that the role of graduate students within the university will drastically change ignores the crucial fact that our role has already changed, a shift that has been demonstrated countless times in the academic literature on the bureaucratization of the university and the rise of the administrative class. The anthropologists Marilyn Strathern and David Graeber, among many others, have written extensively about this phenomenon.
The NLRB decision simply recognizes that a relationship exists between student employees and the powerful administrations of US colleges and universities. To suggest that my relationship with my supervisor, advisors and mentors is even remotely similar to my relationship with the group of highly paid university administrators who determine and regulate the conditions of my employment at Yale (including the conditions under which I have access to healthcare, a stipend, research funding, etc.) is intellectually dishonest and shameful, especially in an academic setting. While this isn't my area of expertise, a quick search on Google Scholar reveals a strong relationship between mental health and precarious labor (e.g., this article). We can tentatively conclude, then, that contrary to Salovey and others' claim that the relationship between students and their professors will be changed for the worse, improved working conditions for students will actually change that relationship for the better.
Kudos to the NLRB for this important decision.