> So what can possibly cause warming is only the effect of the difference in temperatures between the CO2 gas and the surface
This is wrong. If you do the math, you actually find that you get warming from CO2 if the atmosphere is colder, the same temp, or hotter than the surface.
When the ground emits a photon there are two possibilities. it either gets absorbed by an atom in the atmosphere or it goes to space. about half of the photons absorbed by the atmosphere get radiated back to the earth (compared to roughly zero percent of the ones that go to space).
Since CO2 is better than air at absorbing infrared radiation, it traps some energy that would otherwise be lost. The situation isn't really any different than wearing a coat. a jacket will warm you even if the jacket is colder than you are because it prevents some heat from leaving the system. this doesn't violate thermodynamics because the jacket itself isn't causing warming, it's just preventing heat loss to a colder source.
The jacket is preventing heat loss via convection. You literally can't prevent heat loss from radiation. By the zeroth law, an object with specific temperature is at thermal equilibrium with photonic heat bath at the same temperature. If you look at the interface between them, whether there's CO2 there or a black body doesn't matter, because you could replace it with photonic heat bath. The comparison to greenhouse (convection) and to jacket (again, thermal convection / conduction) is misleading exactly because it evokes the wrong intuition.
The better comparison is to a black hole. Literally nothing can escape it - not even light - so it is much worse than CO2. Yet it still has a temperature, and it still emits radiation according to its temperature. You cannot absorb more than black body.
The "better than air at absorbing" isn't an arbitrary statement. It comes from extra degrees of freedom in the gas that can absorb the heat. And the difference in absorbing thermal heat between any two systems MUST be a result of their temperature difference, because they must be at equilibrium if they have the same temperature.
If any two systems of the same temperature had a difference in energy emitted by radiation between them, that would violate the second law, and you would create temperature differences between them (heat would flow) and extract work from the temperature difference without needing a colder reservoir.
This is physics misunderstanding. There's no such thing as "equilibrium temperature" between two systems. The zeroth law of thermodynamics is that all you need to know in order to tell whether two systems are at thermodynamic equilibrium with each other is the temperature. If they have the same temperature, they are at equilibrium, period. You don't need to know anything else.
In this case, a wall which only allows transfer of heat via radiation (say, two simple thin transparent walls that only stop particles with vacuum gaps between them) is a diathermal wall.
Climate arguments with laypeople is usually full of things that go against common sense thermodynamic intuition, and then the funny thing is that when you state the common sense thermodynamic intuition, which doesn't fit the narrative, they object to things which are as basic physics as you can get.
I bet that the confidence in climate change takes a nose dive with understanding of physics. It reaches a stable position somewhere of "well we're doing approximations too so without looking into it, maybe their approximations aren't so bad". If you do go looking into it in detail, you're either too invested to go against it, or you go against it and get cancelled / whatever, and then you're faced with a squad of believers who understand nothing but defend it with absolute confidence, with understanding that sounds more like caloric or phlogiston theory than actual modern physics.
There's a funny social gradient from climate modelers which know exactly how far away from the truth these models are, but still do the best that they can, to their managers, press releases, activists, politicians, where the entire thing looks like a game of broken telephone. And the honest people who say "I'm doing the best I can but I know it's not enough" are probably drowning in the shadow of those with dishonest conviction.
EDIT: to also correct a little simplification, you can prevent heat loss to radiation by reflectivity. But that's not the case with CO2 in the atmosphere, the arguments of greenhouse effect is not about reflectivity. Reflectivity would be the albedo of the surface, which is indeed important. But the atmosphere is completely negligible for reflectivity purposes.
Actually the model for the greenhouse effect is pretty simple, climate models are much more sophisticated than that for example CESM have about 5000 equations as the model takes into account interactions between the biosphere and the atmosphere, clouds and carbon stocks. But greenhouse effects is a really simple you can implement it yourself and verify the results, here's a good start https://en.wikipedia.org/wiki/Idealized_greenhouse_model
> EDIT: to also correct a little simplification, you can prevent heat loss to radiation by reflectivity. But that's not the case with CO2 in the atmosphere
CO2 absorbing some infrared light and transmitting it back to the earth is a form of reflection.
This is wrong. If you do the math, you actually find that you get warming from CO2 if the atmosphere is colder, the same temp, or hotter than the surface.
When the ground emits a photon there are two possibilities. it either gets absorbed by an atom in the atmosphere or it goes to space. about half of the photons absorbed by the atmosphere get radiated back to the earth (compared to roughly zero percent of the ones that go to space).
Since CO2 is better than air at absorbing infrared radiation, it traps some energy that would otherwise be lost. The situation isn't really any different than wearing a coat. a jacket will warm you even if the jacket is colder than you are because it prevents some heat from leaving the system. this doesn't violate thermodynamics because the jacket itself isn't causing warming, it's just preventing heat loss to a colder source.