As bgritzut points out, they always do provide both numbers, but why? Here is how I would explain it: The all-items CPI, as you suggest, is best for measuring the cost of living, that is, the actual impact on people's lives. In fact, for that purpose, it is good to look not only at the seasonally adjusted all-items CPI, but also the unadjusted number (all of these and lots more are given in the BLS press release, to which a link is provided). The core CPI, seasonally adjusted, is better as a measure of the impact of US fiscal and monetary policy on the inflation rate. The reason is that food and energy prices are set in world markets over which the Fed or Congress has little influence. Policymakers can better understand whether their policy is too tight or too loose if they look at the core.
Thanks, I do get that. However, what I am arguing is although the core CPI is a better measure of the Feds influence, that influence might be over an economy that represents a shrinking number of Americans. You can keep the 'economy healthy', but the 'economy' that you are keeping healthy might be shifting towards a subset of the population. For example, GDP can be great, but be due in large part to 40% of the population, or be great and be in large part due to 90% of the population. The core CPI might be something that can be better pushed and pulled by the Fed, however, the real question is how much that matters to the majority of Americans, for which the Fed works for. It could be argued that as disparity grows, the core CPI might be more easily manipulated, and yet less meaningful to the broader population. What confuses the CPI even more, is that those components that are included in the CPI are weighted by the Bureau of Labor Statistics, and they don't necessarily reflect the changing importance of the components. As a college education has become more necessary to get a good wage, and as healthcare costs have become more likely to be able to cause bankruptcy, it's tough to say that inflation has remained low for a family buying heathcare out of pocket and sending kids to college over the last decade.
Your right about the components. The average doesn't really mean a lot. The basket of goods consumed is different for age groups, income groups. There can be effective inflation for one person and deflation for another. Each of us needs our own CPI, really, when you get right down to it.