Let's first clarify some concepts. Free fatty acids, including palmitic acid, are not present in animal tissues (or in the diet) to any large extent; they are esterified with glycerol to from triglycerides (fat), which is the storage form. This is a very important distinction, because triglycerides are chemically inert molecules that can be stored in very large amounts in cells without causing problems (adipocytes are pretty much just one big fat droplet), while free fatty acids are dangerous to organisms because they disrupt cell membranes --- fatty acids are basically soap.
When nutrition studies like the WHO report talk about dietary intake of "palmitic acid", they really mean intake of triglycerides that contain a large proportion of esterified palmitic acid as side chains. There is a large body of epidemiological studies indicating that intake of fats rich in palmitic acid correlates with cardiovascular disease and other "metabolic" disorders, although some reports show no significant effect. In clinical trials, replacing saturated with unsaturated fat seems to provide some health benefit; but there also are studies indicating that monounsaturated fat can be worse than saturated fat. So there is some evidence that too much saturated fat is problematic, although the issue is not straightforward.
The causal mechanism is to my knowledge still not clear. You are right that cells are normally perfectly capable of handling palmitic acid; it is a good substrate for beta-oxidation and a common fuel source for human cells. However, in pathological situations, free fatty acids can accumulate in cells and in the blood, probably because of excessive lipolysis. Many studies show that this can cause insulin resistance and trigger inflammatory signals, and even cell death by apoptosis. Some reports show that palmitic acid in particular causes these effects, whereas oleic acid (monounsaturated) does not, but I don't think there's any consensus on why palmitic acid is special.
My own hypothesis (if I may be so bold :) is that palmitate only appears to be special because cells have evolved to sense it as a "proxy" signal for free fatty acids in general. Since fatty acids are clearly dangerous to cells, sensing them is important, and because palmitate is the naturally most abundant fatty acid, it would be the easiest fatty acid to detect. It is known that palmitate is recognized by toll-like receptors, in particular TLR2 and TLR4, and that sensing by these receptors trigger "proinflammatory" signalling. But this alone doesn't prove my "proxy" hypothesis, of course.