Who thinks this? You're talking to straw-men. The latent structure is absolutely present in the data, but a series of transforms makes the structure more readily available. (Where else would it come from? Forest spirits whispering to the model?)
The 'availability' is made evident by training a linear regression for classification on the learned representation (keeping the base network fixed). This works decently-well for a good representation, and not at all for raw pixels. The point is that you can use 'simple' algorithms - including kNN - on this transformed view of the inputs; it's not magical at all.
FWIW, I find the classification problem space to be riddled with conceptual holes, while other application areas are much less problematic. The use of neural networks for compression makes a lot of sense, and much of representation learning can be thought of as augmentation-invariant compression for some family of augmentations. Some of the recent big advances are really about getting away from the problems of classification.
Meanwhile, reinforcement learning, diffusion learning, recurrent networks, and GANs all bring different kinds of 'interactivity' into the learning problem, and resolve your complaint about ML algorithms working on 'just data.' The iterative processes in these models provide space to push against the data with decisions, and react accordingly.
I find your complaints to be lacking in curiosity or even knowledge about what people are actually doing in the field... Great results keep piling up and ML systems are used by billions of people every day, while you continue to call no-true-scotsman...