Other already gave good answers. The main part is that a house is not a consumption product - it is not "consumed" (used up) in a limited timespan, but rather, a bundle of a long-durable part (building) and an ultra-durable asset (land). Pretty much all durable assets have increased in price since the 1980s in tandem with falling interest rates[1]. The cause of the long term decline of interest rates is largely unknown.

The BLS largely circumvents this by using rents, actual rents for renters, and owner equivalent rents (OER) for owners. This is done by asking owners what they think their house would rent for, and using those increases for the housing/shelter component of CPI. Rents (which make up 1/3 of CPI) have increased faster than the general CPI, but not as much as house prices[1], likely because of the interest rate decrease.

There are some other complicating aspects around house prices (city prices increased faster than rural, houses sizes grew while household sizes shrank[3], so part of higher prices is just people buying more). But I believe the main aspects is really falling interests rates. A proper decomposition and attribution of most aspects probably takes months of work, enough for a econ Master theses. That's why I mentioned I don't have the energy for that. Nor do most other bloggers/pop-article writers, so they just go for popular appeal and clicks, by telling you why everything is getting worse and more expensive for you.

[1] https://fred.stlouisfed.org/series/MORTGAGE30US [2] https://fred.stlouisfed.org/graph/fredgraph.png?g=BLyd [3] https://azgolfhomes.com/average-home-size-u-s/