Nope. CPI is an excellent differential indicator -- "how much did a typical person's cost of living rise this year" -- but it's a terrible integral indicator if you compound it because it's blind to the difference between forced and voluntary substitution. If essentials inflate faster than wages, money_in=money_out drives a reduction in nonessentials -- forced substitution -- and the CPI basket adjustments launder the forced substitution into voluntary substitution.
Well, "launder" is a strong word that the hardworking bureaucrats at BLS do not deserve, but the people who use CPI as a deflater so that they can wave around graphs "proving" that things have never been better absolutely deserve it, so I'll keep it in.
Bonus meme: the American Dream was not to Owner Imputed Rent a house.
Yes, it also takes into account rising quality. For an example, in 2010 I rented a rat hole apartment for $x from a fisherman who had inherited the building. He never did maintenance (he was out to sea most of the year) and he never raised rent.
A large company bought the building after I moved out. Ten years later, the same apartment with a fresh coat of paint and new countertops was back on the market for a rent of about three times $x.
The CPI can say that apartment, since it was refurbished, increased in quality and so it wasn't really a price increase of the same good from $x to $3x. This offers a "degree of freedom" to adjust the CPI itself (since quality is inherently subjective), and may be a big part of why CPI does not reflect the lived experience.
I didn't care one bit about paint or countertops when I rented that apartment and I assume broke young adults today don't either. At the time I wanted the cheapest place to live in the area and this was it. It still is one of the cheapest places, but you need three times as much money to rent it.