These are really complicated systems, and one of the reasons we model them is that they're too complicated for any one person to understand every single aspect; models are a sane way to integrate the results of studies requiring disparate expertise (or at least different people).
I am a modeller, and I simulate the brain a system which is also mind bogglegingly complicated.
The secret ingredient that makes modelling work as an enterprise is the ability to make predictions, and then test them through experimental manipulations. The reasoning is that if your model captures a gem of truth, then it should be able to accurately predict data that you as the experimenter haven't seen yet.
Or at least, that's the theory. The ugly truth is that even in brain science that's a standard which is rarely lived up to.
And the situation is bound to be worse with modelling in climate science. One cannot even perform experiments because we don't have multiple earths to play around with.
So, while modelling is a way for scientists to explore theories and communicate, as you indicate, I fear that the climate modelling process is fundamentally bankrupt because it's impossible to run experiments for the purpose of testing models.
They do make pretty pictures though. A video of a virtual earth turning from blue to orange is extremely compelling!