What’s a summer blue daytime sky?
Anyway, interesting calculation. The wonderful book “Sunsets, twilights and evening skies” by Aden and Marjorie Meinel contains the attached plot. It indicates a factor of 70 million between a perfect night sky and the noon zenith sky. That is 2.5 * log10(70E6) = 19.6 mags. Assuming a perfect night sky to be 22 mags per square arcsec that would put the noon daytime sky at about 2.4 mags per square arcsec so a bit brighter than you calculated.
The surface brightness of the Full Moon is around 3.4 mags per arcsec so that would imply that it is about 40% the surface brightness of the daytime sky which would be easily detectable with the naked eye. That is something you should be able to demonstrate easily by taking an image and measuring it.
The cloudless sky is, of course, not white. Neither is it unpolarized. The use of a polarizing filter can make a big difference in its brightness but a more important method of darkening the sky is to observe through a red or near infrared filter.
I have already posted images of 6th and 7th magnitude Pleiads taken not too long after local noon. With my kit I estimate that I should be able to reach 12th magnitude with a Sloan i’ filter. It will be necessary to take numerous subs where the stars are almost but not quite saturated and then remove the almost as bright sky background in software.