Indeed, to support your point, I had a teaching assistantship for my first two years of grad school, meaning that I worked on an as-needed basis for things like supervising labs, proctoring exams, grading papers, etc. It was funded by the department. Where I went to school, the TA workload was pretty light.
Then I got a research assistantship, which was funded by my advisor's grant. I meant that I got paid to do my thesis research, and no longer had to teach. This was a precondition for joining his group.
My department introduced a rule requiring us to TA a certain number of times. There were too many research fellowships and they were short on TAs.
My group was new, lacking much in the way of research funds. Therefore, we taught for support (Chemistry): lab proctor, recitation leader. Somebody had to wrangle those 1000 people taking 1st-year chemistry. Pre-med, pre-vet, pre-dental, biology and a rare chemistry major or two. In return, the department covered my living expenses. This was in ancient times, but I figure similar mechanisms exist today because the student base seems to be quite similar.