The downforce isn't there to counteract air under the car (actually fast flowing air under the car produces downforce; read up on ground effect and Bernoulli).
The downforce is there so the car can go around corners quicker; the downforce increases the friction (Ff<=mu x Fn), i.e. the force of friction is less than or equal to the coefficient of friction (mu / μ) multiplied by the normal force. In the case of a tire the normal force is the downward force (provided by the wings or spoilers). Therefore the more downward force the higher the friction hence the more the tire can exert lateral force (cornering force). For example a F1 car at high speed can exert approximately 4Gs of lateral force when cornering, or 5Gs of force when breaking (at slower speeds a F1 car can't brake as hard).
Dragsters don't go around corners (not while racing, anyway), but they still benefit from the extra weight on the drive wheels produced by the air hitting the top of the spoiler.
It's the same effect though - the F1 car cornering needs to maintain sufficient friction to prevent the tyres slipping laterally, whereas the F1 car braking or the dragster accelerating needs to maintain sufficient friction to prevent the tyres slipping longitudinally.
The air hitting the top of the spoiler isn't generating anywhere near the suction created on the bottom of the spoiler with air underneath having to speed up (and hence have much lower pressure compared to the top).
Aerospace engineer here: Producing lift in either the up (airplane) or down (spoiler) direction generates induced drag. The amount depends on your design, in that a large real or effective AR can reduce it, but it's going to be there.
I think what happens in automotive designs, which are not always penned with the same attention to aerodynamics as aerospace designs, is that they often have unintentional lift at the rear. Adding a small spoiler, then, doesn't add actual downforce, it just reduces lift -- and therefore reduces drag into the bargain.
They produce downforce to counteract the lift effect of air moving under the car.
It is only necessary at very high speeds, and prevents loss of grip in (especially) the rear tires.
But, yeah, horse and buggy. I'm with you there.