How would that even work?

You get together a group of grad students willing to lead the organization, you hold town halls 'n meetings with students and you talk to them and learn what's making life hard for them and plan how you can improve it. You talk to your union reps (here, the NEA) and eventually get signatures and form a union. Then you cross your fingers that you won't have to go to court like the union at one of your sister universities did.

Grad students are generally in a competitive and underpaid market, aren't in that market for a significant amount of time

Exactly, which gives us so very little leverage and basically no bargaining power. If you don't like what the school decides to do, tough shit, put your head down and try to get out as fast as you can.

Just to put things in perspective, at my school we do not get health insurance or any other kind of benefits despite often working jobs or being on grants that forbid us from working elsewhere at the same time. I don't even know how grad students with kids make ends meet.

To hold a TAship and most RAships you have to be enrolled full-time. It used to be that TAships paid about $16k/yr and then you paid $8k/yr back in tuition. And, let me tell you, the cost of living here is cheap but it sure as hell isn't $8k/yr cheap. A few years ago they finally put in a program like most other schools that reimburses tuition if you hold an appointment, but lately they are trying to walk that program back because it costs money. Last year they kinda jumped the gun and people got mad, but this year they've made some smaller changes that slid by unquestioned and I fear they're just trying to carefully boil the frog.

(And, as a side note, for me that $8k/yr pays for "research credit" which translates to "I talk to my advisor sometimes".)

Being on some research grants isn't much better. I've had a couple dry up with pitiful notice (one literally the Thursday before the semester started), and the one I'm on now penalizes you for having money in the bank, then shrugs when you complain that it takes 2-4 months for them to cut you a stipend check and that travel grants require you to basically loan the university a few thousand for a few months and look it's not like I'm expecting a lavish standard of living but it'd be nice to not have to worry about bills and food when I hold a job that requires some amount of postgraduate education, y'know?

are trying to become the employers that they'd be collectively bargaining against.

For what it's worth, my advisor is vocally in favor of grad student unions.

I'm sure some folks are in grad school to get into higher ed administration, but most are just trying to be professors or R&D industry workers. Professors' hands are often tied with respect to paying their students -- the university sets TA pay and most grants have rules about how money is spent. It's hard for one professor to say, "hey, I need more money to pay my students well"; it's a lot different to be able to work collectively towards raising expectations about grad student pay.

Basically, I just want to be treated like I'm valued a little bit, even before I hold the piece of paper that tells everyone I'm worth something.

(I definitely think that professors' attitudes towards students without degrees contributes to the whole degree inflation thing.)