Surely putting together an inflation index consciously designed not to replicate an average American diet would be a much, much worse measure of food inflation than one that does actually try to model the average American's situation? "Everyone should be going paleo so the index shouldn't have bread in it" is a self-evidently bad idea, that's not a statistician's job. A single inflation number is hardly ever going to be a correct measurement of any single person's experience of inflation, but that's inevitable.
There are real criticisms to make, such as the average basket of goods not characterizing the inflation felt by the poorest, because they spend money on different things than people with a bit more money.
It is just an (imperfect!) aggregate measure of prices, and it certainly seems to capture something- inflation was low for a long time, then COVID, then it got high, then rates went up, then it went down again. When inflation is very high there aren't many inflation truthers going around insisting that actually, inflation is low; when inflation falls or stays low, there's always people insisting that actually not everything is great. Which: yes! You can have low inflation and all sorts of problems! But having those problems and high inflation is worse.
It's not hiding anything, it's just not rich enough to capture everything you're interested in.