Your proposal is unfeasible. How do you propose to get the needed air in? If you hide the intake and cooling openings underneath you would not have control of aero in the underbody the way you do now, you would be inviolation of the technical regs, and introduce a number of logistical and engineering nightmares and headaches.
I see your reason, but how come in the late 80s the cars didn't have the vents above the drivers heads? It shows there are other ways of getting round the problem.
Also back then some of the cars were turbocharged so they would've produced a lot more heat [I'm guessing]
Also I made the mistake of thinking that heat from the engines could simply be blown off without needing intakes, so I'll give you that. However if the gills were made of a heat conducting material , then the heat could be dissipated like a heatsink does in a computer. Also the low pressure produced from the air moving over it would pull the air out, but that would still need an intake of some sort...
That sort of solution might produce less air resistance than the current aerodynamic shape though! But still, You have good points, and thanks for reminding me you need an intake as well as vents to have an to have an airflow!