Interesting...
I know all of these things are relative on field, school, PI etc. etc. But a condition of getting departmental funding similar to the GRF at UC Davis's Plant Science department is that you TA for at least one quarter as a Masters student and at least two quarters as a PhD student. Some might think this offsets the cost to the department, and it does, but the department chair is a big proponent of students TA'ing per the fact that all faculty in our department have dual research and teaching appointments. Apparently the lack of teaching experience has been a deal breaker in job candidates being offered positions here as well. Most of the professors I know also are proponents of their students TA'ing because they suggest that it actually makes graduate students better researchers because they have to answer critical questions from undergraduates which can really help develop our research questions. I would also suggest that TA'ing after the first few years would be much more manageable as most of the required for a graduate students program would likely be out of the way only leaving research and TA'ing a bit. Lastly, if need to take qualifying exams, there probably isn't a better way to prepare oneself for those than if you have to teach the material.
You are correct though, it isn't easy and there are plenty of reasons to argue that we shouldn't have to do it all. I hope I get the GRF but even if I do, I would likely still TA or at least help organize a few seminars. Good luck!