At least some of the objections seem to be that low cholesterol is a common side effect of being very ill, and that lifelong low cholesterol is healthy.
This sounds as though an unexplained drop in cholesterol should be used as an indicator that a person is in trouble, just as an unexplained drop in weight should be, and also that longitudinal studies of individual health and cholesterol should be done.
Discussion of a Norwegian study looking at 50,000 people who didn't have pre-existing heart disease for ten years. http://drmalcolmkendrick.files.wordpress.com/2012/09/mortality-and-cholesterol1.png?w=600&h=309
Here's the study.
Here's the actual conclusion from the study, which dhoe pointed out in comments is considerably milder than the quote above:
However, the chart (the png link above-- I don't know how to make the image appear) shows that the all cause mortality for women was lower if their cholesterol results were higher.
A different big study which also found that low cholesterol was dangerous, but high cholesterol was also dangerous in terms of heart attacks, though mostly for men under fifty, and (I think) not so much for women.
A comment explains that the usual test for cholesterol isn't actually for cholesterol, it's for the lipoproteins which keep all sorts of fat molecules from forming large blobs in a watery environment.
This sort of thing appeals to a number of my prejudices, so I'm hoping to get some more meticulous angles on it from LW.
Post edited to add discussion of the conclusion of the Norwegian study.