Have to admit to being prejudiced, being the Senior Associate at ERI who publishes both labor cost data and cost of living data, so you can ignore me it you want. Your Help menu contains myriad short explanations of the differences between what people earn (salary) and the cost of what what they buy (usually a "family" cost of living).
There are additional articles and white papers on the subject, including one originally called Cost of Living 101. To summarize, people usually don't work within walking distance of home. They live one place and work in another. They live on (and spend) a different sum than their single salary would suggest; and fund it by a combined budget (multi-earner families, credit card and home equity debt, spend savings, inheritances, etc.), and such. The two concepts are apples and oranges.
The politicizing of minimum wages has finally produced a tiny amount of positive statistical significance to living costs today at the bottom levels, but only as it affects the entry salaries in the majority of states with overrides to the Federal minimum rate. Once you get up into the high 5-figure level, the relationship of local living costs to pay disappears. Salary surveys, however, will always pick up whatever real impact there is from family living costs. If costs spur people to demand higher pay, then the salary survey will show it. Using both salary surveys and COL surveys would give the employee a double-snapshot, if there is a relationship. And if there isn't, then forget it.
BTW, I do know that some consultants have advised clients to do what you described: to pay both the competitive market premium pay rate AND add something extra for "cost of living", but I consider it a bogus cop-out. It pleases the employees and it sure makes it easy to hire new people, though, when you pay that much more than anyone else in town. So I understand why they may consider it smart consulting practice. If the outfit that follows that bum advice can afford to stay in business while others get the same productivity from lower labor costs, of course.