The cover only really works at highway speeds where your car's high and low pressure areas really move the air.
Try this test. Turn your AC and Heat off. Fan off. Then drive with the windows up at about 75mph. Notice there is still air blowing in your vents? This air is coming from the high pressure area at the base of your windshield where the HVAC system draws in fresh air.
Some of the same ideas are used to help move air through the radiator/condenser and FMIC. The engine bay is a low pressure area, and the grille is high pressure.
As Keith noted, the high pressure air that goes into your grille, would just dump down mostly under the car if not for the cover. With the cover there, it is drawn through the core support area (Where your radiator/FMIC and other coolers are.) and then dumped under the car into another area of low pressure near the transmisison.
To have a properly operating cooling system, you need the undertray, and a good fan shroud. There are many people who have run the Supra with neither a undercover or a fan shroud, but at speed, or sitting in traffic with the AC on, they will start to overheat the car. (Might be why so many MK3's come down with the BHG? Don't know, but It does not help.)
The system is so effective, that most older MK3's have leaves, pet hair, and other debris packed up against the radiator. This crap was sucked into it as the air was drawn through the radiator core by this pressure change. (And the obstruction on older cars is another reason why so many have overheated in the past and wiped out head gaskets too IMHO.)
Best Idea I've seen yet was using plexiglass to replace lost or damaged undertray's. (Or just get another stock one.)
Helps with downforce too, but that's another post.