Quantify eventually. What temperature do you expect the internal wall surface to be during the heat of the day, on a robustly insulated cavity wall?
You wanted graphs? This is my home's temperature graphs for the past 24 hours. The yellow represents the indoor temperature, the blue the outdoor.
My too hot weather regime, is one of closing windows, drawing blinds to limit solar gain, just having a fly screened back door open for ventilation.
My home is reasonably well insulated, hence the yellow line is fairly flat, but it does drift slowly up, as the outdoor temperature increases. Come bedtime, at around 11pm, the windows are opened to vent the heat gain (despite the insulation) from the day/human activity/use of appliances.
You will note - insulation can delay temperature rises, and falls, but it can only delay the inevitable, and smooth out the average.

