The arguments about grad students teaching classes is an old one, that professors don’t want to be bothered teaching undergraduate courses, they want to spend their time on research and writing (which is not unsurprising given the ‘publish or perish’ mentalithy of many schools, where a teacher who is a great teacher but doesn’t publish books and articles and such that much, is looked down on). This is especially true in introductory courses, where you often do see grad students, or things like core courses for non majors and the like.
However, there are places where grad students have always taught, things like lab lectures, recitations and the like are often taught by grad students whereas the main lecture could be taught by a full professor of some sort (I say some sort, could be an assistant professor or an adjunct teaching, these days that is a lot more common than tenured faculty especially in lower level courses). The reality is that schools often hype their ‘great teachers’, but said teacher often teaches only a very few classes…but the regular profesors/adjuncts/assistants do teach a lot of the classes, even back in the dark ages where rage against grad students teaching courses was a big topic of conversation, a lot of my classes were taught by full time faculty.
One note about TAs, be careful in music about the use of the term, often music professors have teaching assistants, but they aren’t like a typical grad TA, many of these are fully qualified teachers working with a ‘master teacher’, the way for example Cathy Cho at Juilliard works with Perlman (or did, not sure if she is still his assistant), Delay was Galamians assistant there, and there are a ton of high level teachers who once were Delay’s assistants.