Airplane Shell
Let's do a thought experiment: what if we increase the number of planes in the air until they blot out the sky1? If pressure transmitted to the ground is not holding them up, then what is? Does the air itself have some kind of magical inertia that allows it to suspend planes without themselves transmitting the force? Does the density of this magical air matter? Could we thin out the atmosphere and still maintain this "airplane shell"?
Let's do just that. Let's put all the planes on the ground, and cover the surface with runway, so the planes are rolling around on their wheels. Do we agree that the ground is holding them up? Now, let's gradually add air until they lift off. Note that as their speed increases and they generate lift, the suspension will hold up less and less of the aircraft weight. Where does this lift come from, and what is it pushing against? The wheels are pushing against the ground, so if the air isn't also pushing against the ground, you need to invoke some magic to tell us what the air is doing differently than the wheels to counteract gravity.
Ground Effect
Now, it turns out that the lift near the surface does something special, called ground effect. This increase in lift is so pronounced that entire vehicles have been built to take advantage of it. One of the largest such "planes" was the Russian-built ekranoplan. @Dale's dramatic picture of an F-18 skimming the ocean looks awesome, but doesn't directly imply ground effect, because we know these jets can fly at tens of thousands of feet. Whereas, the ekranoplan couldn't even get 100' into the air, because it is completely dependent on ground effect, which is the increased lift from air pushing against a nearby surface (like the ground, usually).
Escape
Now, if you got a big metal plate and tried to float it in a swimming pool, if the plate exactly fit the surface of the pool, then the water should be able to hold it up. But what if the plate didn't exactly fit the shape of the pool? Or what if it had a hole in it? Well, obviously, it would sink, and water would spurt up every opening it could find. 1018
Huh.
So let's think about this for a minute. Gravity is pulling the plate down, along with the water. And the water molecules are kind bouncing around near each other mostly not going too far2. But when the plate-with-a-hole begins to sink, we don't just have metal pushing on water pushing on ground. We also have metal pushing on water, pushing on a water jet that shoots upwards, opposite of the plate's direction of travel!
Now technically, the falling metal plate is not directly accelerating any water molecules upwards. Rather, the plate is increasing pressure on the water beyond the gravitational weight, and the water molecules simply escape through whatever free path they can find. The fact that liquids don't hold their shape allows them to convert the downward force of the plate into an upward force on some of the water molecules. So, even though most of the weight of the plate is being transmitted by the water to the bottom of the pool, some of it is actually causing water to escape from the pool.
In the same way, if our "airplane shell" were "floating" on our atmosphere, but were not air-tight, then some of the lift would actually go towards forcing the air between the cracks, and accelerating it away from earth. So even though most of the force must eventually get to the ground to support the weight of the airplanes, the slippery nature of gases ensures that we cannot direct all of the lift exactly downwards.
Note: all of the weight of the airplanes must be ultimately transferred to the ground, or they will not be able to maintain level flight. However, the "weight" of the airplane also causes forces to be transmitted in all other directions as well, including upwards.
Notes
1 Let's ignore the fact that there isn't a feasible flight path for all those planes, assuming straight and level flight, or that they don't perfectly tile the surface of a sphere, or that aerodynamic effects would prevent them from flying so close together, etc. Just pretend they are spherical cows.
2 If they were going far, you would notice them as currents, although, diffusion of dye in the pool should show you that a single molecule could make it all the way across the pool eventually, just from Brownian motion. 461