As to the virtue of the green house effect studied on its own, the issue is that the atmosphere just might not work that way.
And that's why we have more detailed models.
But it ends up being a no-win situation, when the objective is to criticize rather than to understand. If the model is simple, they say "that model's too simple! The real world is complicated! You need to include X Y and Z!". And if the model is modified to include X, Y, and Z, they say "The model is too complicated! You can't believe any complicated models like that!"
The actual answer is, you start simple, understand the simple models, and progressively add complexity. This is the way science is done. Planets don't move in uniform elliptical orbits. Nevertheless, starting with Kepler's laws and then adding peturbations is a good way to analyze planetary motions.
As for comparing models to reality, and asking what we know and how we know it, there isn't really time for me to go through this model by model since 1967 (since I do have other things to do). I'll again suggest as a start reading the WG-1 summary report, it goes into detail on this (and has references for more details). The one I'm more familiar with is the fourth: https://www.ipcc-wg1.unibe.ch/... (although there seems to be a more recent one, fifth, here: