I believe the hens DO stop turning their eggs and most do not get off the nest in the last few days. Sitting tight probably does increase humidity, though I have obviously not stuck a hygrometer under any of my hens, lol. The chicks hatching and drying would definitely raise the humidity, as it does in the incubator.
My thoughts on humidity is that their is a lot more area in the bator to humidify than there is under a hen. My drying chicks might raise the humidity 5% in the bator, but will not bring it from the 35% that I incubate at to the 65%+ I like to hatch at. Lower humidity and my chicks definitely stick in their shells. I have had a lot of problems with hatches at lower humidities.
A lot of people want to try to compare what we do in an incubator to what happens under the hen, but I think we need to remember that there is nothing natural about artificial incubation.