That would have been true if CPI was a purely economic issue. But it’s also a political issue and once politics is involved, there is vast incentive to fudge and massage the data to fit a narrative.
It is probably the most looked at number that the government releases. There are economics professors/researchers, statistics professors/researchers, economists at hedge fund, economists at pension fund, economists at labour unions, bond traders, equities traders, etc, looking at the numbers every month. No one professionally involved with it thinks it's fudged.
There are certainly different methodologies, and that is often debated (e.g., how best to measure Shelter for homeowners versus renters), but for the current system, I'd like to see which non-tin foil hat wearing people (i.e., not Shadowstats.com) think it is being fudged and how.
And? It has always sucked to be poor / have less. That doesn't mean that the measurement of prices via CPI is inaccurate.
The fact that capitalism has run amok and income inequality is growing doesn't mean the statisticians at the BLS are wrong or are cooking the books.