Preference utilitarian measure of historical welfare

I have a question about how this would work that is related to the both the values and representative sampling issues you raise. Would I or would I not adjust my gender (female) in placing my bets? Would I assume that I would be a woman of around median status and income when compared to other women of the era, or when compared to the overall population? It practically goes without saying that in many of these eras, most women had low status compared to men. Even if the more important factor in determining your position would be income, many women in many of these eras would only have income/property as a member of a household and in relation to men.

Somewhat relatedly, would someone of African descent assume he or she would be of the same ethnicity if rating, for example, the antebellum South of the United States (or many other eras/locales in the U.S.?)

It seems to me that in many cases the values problem you identify isn't just about what one would find morally repulsive even if one's physical welfare was not so bad. In many eras and locales, a women or some particular ethnic minority would have a very different lot in life in a way that directly affects his or her welfare (e.g., as a woman, not having any property rights and thus lacking security and independence; as an ethnic minority, being oppressed or even a slave).

Should everyone making these ratings take that sort of thing into account (that is, the possibility of ending up being a woman or an oppressed ethnic minority in the new era), or only raters who are female or of an ethnic minority? Did you take this into account in making your own ratings? It seems like that sort of thing could greatly affect how one rated different eras.

There's evidence that happiness is proportional to the log of income (blog post, pdf article) That suggests that GDP per capita is a decent measure of quality of life, we just shouldn't treat it as a linear relationship. Exponentially increasing GDP translates into linearly increasing happiness.

Log income predicts happiness at a given point in time, not for a whole life, so I'd expect your method to produce ratings that are closely correlated with log income multiplied by life expectancy at age 20 (to match your decision to exclude child mortality). If we can find historical data on life expectancy and GDP per capita then we could test that prediction.

Being transported from 2010 to 1700 isn't the same as being born in 1700.

Your formulation of the question sounds unintuitive to me. We could ask a simpler question: would you rather live one life starting 2010, or two lives starting 1700?

Also you could try applying your technique to comparing the welfare of different countries right now. Many of the problems you listed will be easier to overcome.

Neolithic Middle East (5000 BCE) - 1.6, Paleolithic anywhere (20000 BCE) - 0

Why do you prefer neolithic Middle East to the paleolithic? The advent of agriculture probably decreased the life expectancy, and I don't see anything which could compensate it in early agricultural societies. Put another way, I would strongly prefer a dangerous, but at least a bit adventurous life of a hunter-gatherer to the slavish work of a primitive peasant.

This was proposed as an alternative to GDP, but it's not clear that it actually measures something similar. Even broadly understanding both as attempts to measure human happiness, it doesn't seem similar.

Since we have no access to time-machines, we cannot give anyone a real choice between travelling back to 1700 and staying in 2010. There are no actual consequences to what they choose. So we are not even measuring people's naive preferences, we are just measuring what they like to say or believe about 1700 vs 2010.

If we're unhappy with GDP there are alternative measures like HDI, but they're highly artificial. It would be very easy to construct completely different measures which would "feel" about as right.

Running with this side point: I wonder if it would be possible to invent a less arbitrary HDI by feeding the variables the HDI uses (life expectancy, literacy rate, educational enrolment, and log GDP/capita) into a PCA and using the first principal component as an HDI replacement. That'd be less arbitrary than the current average of arbitrary indices method.

This makes sense to me, but my ratings would be very different from yours. Also, is your rating for Western Europe 1900 colored by hindsight of two world wars, viral encephalitis, spanish flu and the rise to domination of the bureaucratic state? How clear are you being about socio-economic class? Are we just assuming the population distributions that existed? If so, ancient world slavery might make it less appealing than the Paleolithic I think.
As noted, time travel is problematic, but in what sense could a Paleolithic person 'be' me.