The upshot is that the above mechanism doesn’t do anything to explain why the ground temperature is what it is. In reality, it doesn’t even explain why the atmosphere has a temperature gradient. The primary reason for the temperature gradient is the simple fact that the ground is hot and deep space is cold - enclose the Earth in a high-tech sphere that exactly matches the ground temperature at all times, and the atmosphere would eventually settle to a constant temperature from top to bottom.
What it does explain is convection: when a packet of air warms due to contact with the ground, what happens? Well, the pressure is still the same, so according to the ideal gas law its volume must increase, driving its density down. Being lighter than the surrounding, cooler air it therefore rises due to buoyancy. As it rises, it cools. But, since this is the real world where it’s not enclosed in a magical perfect-insulation balloon, it cools by more than what the ideal gas law dictates as it gives up energy by radiative, conductive and convective heat transfer to its surroundings. So eventually it stops rising, gets pushed aside by the air still rising from below, and starts to fall again.
I'm honestly not sure if I can make it any simpler for you, but here goes: when you have something hot on one side, and something cold on the other side, and something in between, the "something in between" will have a temperature gradient across it. Here, the ground is the "something hot", deep space is the "something cold" and the atmosphere is the "something in between". The gradient appears because heat is constantly flowing from hot to cold.
Does this negate the fact that air expands and cools as it rises against gravity? Not in the slightest. What's going on here is that you're hopelessly mixing up cause and effect. In my fictional scenario of a perfect shield around the Earth maintaining the same temperature as the ground there would be no temperature gradient in the atmosphere. How can this be reconciled with the fact that rising air cools? Simple: air would no longer rise. There would be no convection, no wind. Just eerie stillness. Because the driving force that causes air to rise and cool in the first place is the fact that air at ground level is being warmed by the ground (which is warmed by the sun) and the air in the upper atmosphere is being cooled by heat loss to space. All of which I explained in my previous post - which you failed to understand, just as you'll fail to understand this one.
