The biggest heat transmission problem in most construction isn't the walls, it's the windows. The amount of heat transmission you get via glass - even high-performance triple-pane glass - is significantly higher than heat transmission through the rest of the structure. Your walls are going to be R20+, your attic insulation will be R35+. Your best windows are going to be equivalent to R6 or so, with the median windows being significantly lower.
I think that the best combination is high thermal mass structure followed by the insulation inside. This way the heat is not easily transmitted to the inside and the thermal mass can cool off to the outside during the night.
Of course when the heat finally does get inside then it is trapped because the house will act like a thermos and only way of cooling would be the by circulation of cold air.
Problem of windows can be solved by outside blinds (or shutters but blinds are better) and IR reflective coating.
[1] I don't recall the other design details that made it so cool in temperature, but it was all analog/natural (i.e. no AC)
As you noted, this used to be a fairly standard design for homes in some areas.
Perhaps a "whole house attic fan" could help on those marginal days?
The same thing happens here in Arizona; the proper way they should be building houses here would probably be a monolithic dome with most of the home set below grade - although caliche would make that difficult in some areas (and it would be expensive regardless). The next best thing would probably be rammed earth with 2-3 foot thick walls.