The BLS goes into excruciating detail about their methodology, which is not something often find scams doing.
The Consumer Price Index is just that, an index. It seeks to distill the complexities of the consumer economy into a single number, and to do so reliably over time. It's impossible to do this without a whole bunch of compromises and trade offs. It's trivial to rip apart any such methodology because of these trade offs - just find a trade off you don't agree with and argue it should be the opposite.
If someone attempted to make a t-shirt using the weighted dimensions of every American, the result would be a t-shirt size that would not fit any individual. But, over time, that t-shirt could be used to understand changes to the American physiology over time.
> For example, they attention to adjust the new price to account for what they consider to be added value.
They cover this in their FAQ [1]. But again, this is a trade off. If laundry detergent becomes more concentrated, thus can do 20% more loads for 10% more price, then how should that be handled? If $1000 Nike shoes become popular, does that mean that the price of shoes increased 10x even if cheaper alternatives are still the same price?
[1] https://www.bls.gov/cpi/quality-adjustment/questions-and-ans...