Any ideas of something specific that changed, either between 4.4 and 4.8 (kernel startup code, default framebuffer modes, etc ?), or between the 340.96 and the 375.66 drivers ? In any case, I'll try that specific combination of kernels/drivers home to see if I can get it to a useable state.
Update, July 23rd
I reverted to the earlier drivers/kernel, to no avail. But it seems in fact the problem with my work computer is linked to an allergy of some kind, since antihistamine drugs have an effect (there are construction works in my lab, perhaps they are the cause ?). No idea still for my home computer, for which the problem is definitely not solved.