That's not exactly how it works.
The faculty receives a funding, from a public source such as NSF/NIH, or private source, maybe Google/Facebook, or donation. Roughly 50% of that goes to the the university in lieu of overhead, admin, facilities etc. expense (this should sound an alarm bell to anybody reading). Roughly 25% covers tuition expense- again to the university. The remainder is that $34k you are reading about.
So yes, the NIH/NSF will grant more than $100k of the taxpayer money for one student, but the university takes 75% or more of that and peanuts are left for the grad student workers. If someone is in a good place, they might have $1500-2000 take home every month. If a they are in a bad place, it could be lower than $1000 a month. Basically you'd make make less than half of what a fast food worker makes. But without you, grading, research and all other critical higher education activity will stall.