The index doesn't include housing, food, and energy either as those are considered 'too volatile'. It's no longer targeting what anybody actually consumes anymore
I don't know where you heard this, but it is not true. The CPI tracks rent/housing, food, energy, tuition, medical expenses, and everything else that people seem to think are excluded.
BLS tracks many inflation values, including those for housing, food, and energy, and they have done so for over 100 years.
You're confusing core CPI, which excludes them so analysts can see the difference, and the CPI values as reported generally in the media, which do include housing, food, and energy.
The most reported CPI is CPI-U (and is called CPI in the press), which is the CPI for urban consumers, which includes housing, food, and energy, and most comparable to historical and inflation values from other countries.
Some comments below keep saying that CPI measures things like food and energy prices.
What we really mean is that Fed policy is dictated by the Core PCE rate, which specifically excludes food and energy prices. So sure, the PCE and CPI measures energy and food prices and other volatiles prices, but *FED POLICY is dictated by the CORE rates*, which specifically exclude those volatile components.
So de facto, no one (ie. the Fed) doesn't care about energy or food price inflation.