Stripping out food and energy is not a claim that they do not matter. It is a signal-extraction choice: a spike in petrol prices tells you little about where inflation will be in a year, while shelter and services tell you a lot.
Within core, the useful split is goods versus services, and within services, shelter versus everything else. Shelter is roughly a third of cpi and lags market rents by about a year, which is why core CPI kept printing high long after new-lease rents had cooled.
Example: headline CPI rises 0.1% on the month as petrol falls 4%, while core rises 0.4%. Annualised, core at 0.4% a month is (1.004 to the twelfth power) minus one, about 4.9%, so the soft headline is misleading.
Related: cpi, core-pce, pce-price-index, ppi, disinflation