[yorwba]

That kind of iid assumption could be summarized as "the training data is representative of the data we want to apply the model to", and if it doesn't hold, that's indeed a problem.

But "Data has to be changed and manipulated into i.i.d. form, or the algorithms won't work. How does an independent set of random variables give us a model of the actual dataset which is a very limited representation of the real world?" strongly implies that the data itself should be decomposed into iid variables. While whitening ("manipulating into iid form") is a common preprocessing technique because it's simple and effective, that doesn't mean that learning algorithms wouldn't work without it. They'd just take a bit longer to arrive at the same result.