This is implausible, for two reasons.
First, the postwar US middle class society was not just about income distribution, but also about high overall levels of wealth. To wit, income inequality in the US today is about the same as before the Great Depression. Today, the US is clearly a middle class society. In the 1920s, it wasn't. There was a growing urban middle class, but most urban Americans were working class, and the country was still 40-50% rural - actually rural, centered on farming, and not the exurban life today that people call rural. How do you build a 19c middle class recognizable to today's Americans, when electricity hasn't yet been invented, cities do not have running water, cars have not been invented yet, and of course there's no such thing as household appliances?
Second, even getting the income distribution in the 19c to be the same as today is hard. Inequality was high all over the developed world then - the Gini indices were in the 45-50 range. It took the physical destruction of capital associated with the world wars to bring that down - Piketty's r > g vs. r < g and all that.