I know what you're talking about -- my ugrad work is in biology and bioanthropology, so I don't think I had a single native English-speaking TA.
I think the problem is that they don't pay the TA's enough. I have some friends who TAed (mainly ILR and Hotel classes) as UG's. They only get paid like $8/hour, so there isn't as much competition for the jobs.
But I think students have a knee-jerk reaction against foreign TAs. One of the women I worked with was from South Korea. Her English was beautiful and impeccable. Her father came very close to becoming president -- we are talking about a woman with a high-class education! I ended up with 65 students in my "small" discussion section because everyone fled hers, claiming they couldn't undersand her. ? I'm telling you, she barely had an accent.
This definitely may be true for some people. But as I talk very fast myself, I am usually pretty good at understanding people with accents. I had friends at school from Mexico, South America, Africa, Europe, and the Middle East. I had no trouble understanding any of them, even if they had thick accents.
On the other hand, I have an extremely difficult time understanding Asian (both Far East and South East) accents. I dread calling tech support or customer service and getting an Indian rep, as I usually waste 10 times the actual length of the call repeating myself and asking him to repeat himself.
Also, even if their English is technically good, you have the cultural barrier, which can complicate normal discourse, as terms and phrases used in vernacular English may not be understood by a foreign speaker.