Home prices themselves aren't really used, because they're based off of interest rates, land values, and many other attributes that don't really apply to CPI. Rental costs for all of their faults, are a better measurement of month-to-month or year-to-year inflation.
So you're right. They don't really include "house" prices. But they absolutely include "housing" prices.
My experience with people complaining about CPI is usually simply that the numbers don't match their feeling.
Looking at the description [1] is really interesting:
> The shelter component of the CPI includes two major items: rent of primary residence and owners' equivalent rent (OER). Rent of primary residence, or simply rent, refers to the cost of renting a place on the market. [...] OER refers to the amount of rent that a homeowner would be paid if they were renting out their own home in the market. [...] is calculated by splitting the non-recurring cost of buying a home real estate into rental payments
I got a bit tired reading [2] for more details, as it isn't even my countries CPI...
> The expenditure weight in the CPI market basket for OER is based on the following question that the CE Survey asks of consumers who own their primary residence: “If someone were to rent your home today, how much do you think it would rent for monthly, unfurnished and without utilities?”
Left me a bit unsatisfied, but I'll keep it at that. The followup question would be how much rent prices deviate from housing prices. I am sure that has a lot of data though.
[1] https://en.macromicro.me/charts/24/cpi-house
[2] https://www.bls.gov/cpi/factsheets/owners-equivalent-rent-an...