Sure, but housing prices (nearly everywhere) have risen fast and healthcare costs (nearly everywhere) have risen fast. Education and Child Care have too.

The criticism isn't "inflation should be based on NYC rent." The criticism is "the basket of goods weighs nice-to-haves too highly, and important needs like rent/healthcare/education too low".

So while inflation (as officially defined by the consumer price index) is technically low, "real" inflation (as experienced in impacts to citizens cashflow) is significantly higher across the board.

Even when correctly measured, there will be a lag because there are many people who don't pay the new prices right away, and inflation is averaged over everyone.

Homeowners don't experience higher housing prices unless they decide to buy a new home. (Though they might have to pay higher property taxes.)

Renters might not experience higher rent right away, either due to rent control, or having a landlord that doesn't raise prices as fast as the market.

Your not taking about an inflation index thec, but a cost of living index. And they are not the same thing. There are many reasons the cost of living can change without a change in the currency, such as a change in the preference of goods or technological advancement in a particular field or price controls or regulation.

If you are going say "but the CPI does measure things that way" you are right. It has been well observed that the CPI had been looking more and more like a cost of living index for years (eg, the hedonics changes). This is why the GDP deflator is used more often to gauge inflation now, except it only come out quarterly with the GDP figures.