Yes, I think you may be missing something…. If the central bank is targeting (future) headline inflation correctly, by responding correctly to current headline and core inflation, then neither headline nor core should forecast future headline inflation. This is an immediate implication of the orthogonality of forecast errors wrt the information set under rational expectations on the part of the central bank…. If (for example) you find that current core predicts future headline, that tells you nothing structural. Instead it tells you that the central bank didn't have rational expectations and was making systematic mistakes by not responding strongly enough to core.
This is just one application of a more general theme I've been hammering away at for the last 12 years. If the Bank of Canada is targeting 2% headline inflation at a 2 year horizon, then headline inflation should be unforecastable from everything in the Bank's information set 2 years (or more) prior. If we find headline inflation is forecastable then we have prime facie evidence of systemic mistakes in the Bank's monetary policy reaction function, and can use that evidence to improve its reaction function, in a sort of "learning by past mistakes" method.
Given that it takes 3 or more years for open-market operations to have their full effect on inflation (one year for interest rate changes to affect employment, and then two years or more for employment to affect rates of price increase and price increase expectations) there is useful information about more than the optimality of monetary policy to be found in the correlation between current core and headline inflation and future inflation at horizons of up to three years.