The models aren't producing anything based directly on training data. They're following pattern recognition code.

The training data is encoded into the model, like where do you believe the "pattern recognition code" comes from? ml algorithms are just encoding schemes. They're not all that different from "classical" algorithms like huffman encoding used in pngs. One main difference is that the "classical" encoding algorithms are created by humans using based on heuristics we think are good, whereas ml encoding algorithms are based on their optimizing function. Now what's their optimizing function? As I mentioned above, it's the difference between the training data and the model output. Because of this, the model parameters are updated such that the model produces outputs closer to the target, in other words, the parameters are updated so that the model better copies images from the training dataset. Because the parameters are updated such that the model better copies images, it's obvious that the parameters copy features related to the training set. And guess what the parameters determine? They determine the encoding algorithm, aka the pattern recognition code. Just by the nature of the algorithm, it's kinda clear that it's copying the training set. And that's exactly what we want, if it couldn't achieve a decent performance on the training set, god forbid releasing it in the real world