Culture of British North America (if the ARW was avoided)

That's not really an accurate description of Canada or Brazil. Canada got more white ethnic migration per capita than the US in the 1900s, and Brazil got so much European migration that the largest Italian city in the world is Sao Paulo. The big US-Canada differences are,

1. Canada got much less German migration in the mid-19c. Germans are around 15% of the US population.
2. Canada's English vs. French linguistic identity politics ensures that Anglophone Canadians of mixed ethnic background are more likely to identify as English-Canadian, whereas Americans tend to identify with the most ethnically marked portion of their ancestry.

*Only* more German migration is an extraordinary comment given this is the vast majority of US ethnic migration! I don't know that anyone would either claim that Brazil matched anything like the US in the degree to which the Brazil is ethnically dominated by Portugal vs US English descent fraction, despite a much lower Portuguese population.

"Certainly lower than France" is a bad example, since France had very high inequality until the 1960s.

Yes, France was used as illustrative, as we all know it was relatively high. I wouldn't mind a cite of diverging US income inequality against Europe in the early 20th century, certainly Britain was not an outlier in the mid-20th century - https://ourworldindata.org/income-inequality.

Oh, God, that paper of Swedish surname analysis. I cringe whenever people cite it.

It certainly does not look at relative income, as they're trying to control social status differences for income inequality between society. Yes, they only look at professional status, and political representation is a different one, and a systematic look would be useful - you're welcome to hold that an anecdotal example holds generally, and that political representation differs from professional and educational advantage (despite overlapping democracy), but one Michael Gove as actual leader or John Major later, you're in very troubled waters. Even on the British front bench Left alone, McDonnell's not from a comfortable background, Kier Starmer is not from a comfortable background, Diane Abbott is not from a comfortable background (her dad was a welder, if you like) - they're the Shadow Cabinet! Regardless, it has to be a long run difference to matter, and not something that occasionally holds and occasionally doesn't, and we'll pick and choose which we prefer, since here we're talking about long run cultural divergence.
 
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