(And yes, I've studied lots of economics and understand price deflators, inflation, etc.; the point is, these comparisons don't really make sense over extended periods of time like this.)
Yeah. It's probably useful in something like a magazine article or book to give a sense for how someone in 1800 would have viewed a $20 bill in purchasing power. But you go back a century or so and the basket of goods you're comparing look so much different.
And behaviors are also just a lot different. Most middle class people in the US won't have a cook but they have tons of dining and take-out options. Just to give one example.
Here's my naive/intuition based attempt to guess how it works: If it varies as a continuous function, and small increments are measurable/well-defined (i.e. stuff like TV's and iPhones don't get invented as point-like events enough to screw the continuity), one could think that treading back increment by increment and then taking a limit would yield a valid time series. Then, everything outside that time series would not be indicative of the purchasing power of dollar, but the price of that specific good or service.
Does my reasoning make sense?
That market basket gets changed over time which makes comparisons harder and harder over time because you're buying things that were maybe "typical" for those two points in time but are very different. Imagine telecoms and entertainment costs for a typical 1970 consumer vs. today.
The issue with consumer baskets has become more difficult but only in the recent past where the quality of goods is often hard to compute. But historical data is much more solid because the relevant consumer basket changed relatively little.
It is also interesting to ask: if what you are saying is true, then what do you think economic history is? There are thousands of price series which do what you say is impossible. To a large extent, the data that we have going back to the 15th century (and beyond) is the study of prices and population. All of the products that people consumed then, they consumed now. Saying that shoes have somehow evolved beyond the comprehension of someone living in the 15th century is...odd.
And btw, this makes no difference at all for this question in any functional sense. No-one believes that if you devised some perfect consumer basket that it would show that the dollar actually maintained value...that makes no sense.
Complaining about the change in quality is something that has become very trendy but this is the least relevant part of the problem. It is an issue when you look at certain products (particularly telecoms) but this isn't really relevant for OP.
A pair of shoes would cost something like $1. A slave, $400.
https://babel.hathitrust.org/cgi/pt?id=wu.89071501472&view=1...
If the price of the modern house is 2x the old house, should it be attributed to inflation or a higher intrinsic value?