The problem with the "adjust for inflation" methodology, is that inflation is not equal for all components of the basket. The CPI measures the average inflation of the basket, but if some components have grown significantly differently from others.
In particular, some components of the current household budget were not included in the the original cost estimates. So the "inflation adjusted cost of living" doesn't incorporate those aspects, which is why rebuilding the number from scratch gets a result so wildly different from the inflation adjusted number, instead of being a few percent off due to a different methodology.
In particular, some components of the current household budget were not included in the the original cost estimates. So the "inflation adjusted cost of living" doesn't incorporate those aspects, which is why rebuilding the number from scratch gets a result so wildly different from the inflation adjusted number, instead of being a few percent off due to a different methodology.