Related, I noticed we use a ChainedAssignmentError with these messages to raise a warning, not an exception. Do you think we should change the name of this subclass to ChainedAssignmentWarning as a clearer name

Yeah, it's probably confusing ..
So it was originally an exception (and was then called that way), but then we changed it to a warning because of some false positives that occurred in cython code (#51926). You asked the question at the time (#51926 (comment)), and so the idea is that it really is an error that you should fix (typically, except for those false positives if you are writing cython code), and that was the reasoning to prefer the "stronger" wording about an error. But of course a warning class being called Error is also confusing..