Inflation is complete bullshit because it’s careful to measure a subset of prices and exclude a large range of other prices. Meaningless crap, designed to support government narrative that everything is great, nothing to see here. To have any credibility at all, inflation MUST include rent/mortgage payments and house prices.
Doesn't the CPI include housing? (food and beverages, housing, apparel, transportation, medical care, recreation, education and communication, and other goods and services) Question 10. https://www.bls.gov/cpi/questions-and-answers.htm
RPI takes into account housing costs such as mortgage interest payments, building insurance etc. I'm not sure what USA measure, but the UK shifted to CPI in 2003 to harmonise with EU.
So no, UK/EU/CPI inflation measures basically don't include housing related costs. Certainly not the price of houses, which might be the thrust of OPs point.