Right, we can’t ignore compensation, but there is no “scientific” way of handling it. We must use our judgment. We are in the realm of economic history, not theory.
I don’t think there is a single index to indicate living standards. We have to look at all the relevant data and make a judgment call.
The Sowell quote said that HH income rose slowly because the average number of people in a HH declined. My point is that it’s precisely because of such problems with HH income that economists don’t use it. They use measures standardized for population, like per capita real income and real wages. Some of those statistics indicate stagnation others do not.