It isn't really a sandwich, it is two different exposures, one with the screen and one with the positive. The screen provides the 'ground' for the etching process. What it does is provide microscopically tiny little exposed areas (roughly 20-30% of the total area) in a random pattern on the plate. This gives broad areas of tone the ability to retain ink when the plate is prepared for printing.
What you are proposing can be done, but actually is a process called photo-etching, where a single exposure is made and the density of the dots provide the gray tones. It is somewhat akin to what you would see in a black and white newspaper print where the relative density of the black dots provides the tone. The look is not nearly as smooth as the two step screen/image positive approach.