That is definitely not true for many non-experimental science departments. Many students are TAs and as such care not a bit about grants. When I was getting my PhD I had to teach 4.5 hours/week (often structured as 6 and 3 at alternating semesters), which gave me my stipend and free tuition (and was a useful skill to polish). Maybe I spent another 4-5 hours preparing for classes and grading (we got student graders but I seldom asked them anything as grading was quick enough).
I was paid by the department and it did not matter to me whether my advisor got grants or not; what mattered were research interests and his guiding of research. This was the picture across the department; changing advisers was quick and based on mutual interest, not financials / grants.
I have never heard of a 20 hour TA load. Again, just wondering.
I'd imagine my experience wasn't an uncommon one.
This is just a couple of data points, though. Maybe we were just lucky.
When I did my PhD I worked in Math and CS (plus had friends in Physics with similar observations): many professors on the theoretical side had no need for and were not interested in getting grants. They would get some minor travel money for conference expenses and pay themselves over summer so they would not have to teach then, but even if they got nothing they would be just fine on a base salary. And I am talking about reasonably well known full professors, not some young researcher at the end of a rope (career-wise).
Things are different when you have experimental labs to run. That is when you need external funding and have to do all that is involved in getting it (proposals unlimited, etc.)