I recommend people use many different indicators to get a mental model of inflation, and you will notice that CPI does not represent inflation well. The memes of Arby’s 5 for $5 becoming 4 for $10 are more informative than the CPI numbers. Don’t let the shock at the grocery store wear off – it’s real and painful despite what the news tells you. True inflation would measure the amount of prosperity achieved per hour w…
With inflation, we want to know about the change in value of the dollar only. We don't want to factor in the change in value of houses, food, etc. There is a place for understanding that too, but inflation is not it. Something like a cost of living index is more suitable to that kind of information.
On that note, houses are unquestionably more valuable today. For one, they are twice the size they were 60 years ago. We can all agree that a bigger house has more value than a small house, at least within reason. You can fit more in it, it is more comfortable, etc. and that is valuable. They are also a lot safer. That too is more valuable.
As such, a housing costing x% more today than as compared to some time in the past does not mean that inflation is x%. Again, we only want to know about how the dollar changed in value, not housing. Of course, if you only have numbers, it is impossible to know what share is due to housing being more valuable and what share is the dollar being less valuable, although we can say with great certainty that it is some combination of both.
CPI tries to tess out what is the dollar portion only by looking at a large basket of goods of things people buy frequently. Where you see common moment in the change in price across all items in the basket, that is assumed to be the change in value of the dollar. It's not perfect. There is no perfect metric. You are quite right that one should look at many different models, even though none of them will be perfect either. It is all good information, none of it great, but at some point you have to pick a single number.
On average, it will be close enough.