What you suggest has already been done: train a neural network with the output of BPG or JPEG, and ask to reconstruct the input with just the decompressed pixel

What you suggest has already been done: train a neural network with the output of BPG or JPEG, and ask to reconstruct the input with just the decompressed pixels being available. It definitely is a valid approach but the limitation is that if the network needs some texture-specific information that cannot be extracted from the decoded pixels, it can't really do much. There were approaches where such information was also sent on the side, which yielded better results, of course. The field is wide-open and each approach has its own challenges (e.g., you may need to train one network for quantization level for example if you're going to do restoration).