At the end of the day, you have some things that are a huge component of the CoL for some people (housing, car) that are fairly modest for others who have a paid-off mortgage or live in a LCoL area and either don't own a car or drive a paid-off old one.

Assuming the inflation of those items differ significantly from a broader basket of goods, there's no way to have a single measure that reasonably represents price increases for the two groups.