Using noisy crowdsourced labels from multiple annotators, a deep learning-based end-to-end (E2E) system aims to learn the label correction mechanism and the neural classifier simultaneously. To this end, many E2E systems concatenate the neural classifier with multiple annotator-specific label confusion layers and co-train the two parts in a parameter-coupled manner. The formulated coupled cross-entropy minimization (CCEM)-type criteria are intuitive and work well in practice. Nonetheless, theoretical understanding of the CCEM criterion has been limited. The contribution of this work is twofold: First, performance guarantees of the CCEM criterion are presented. Our analysis reveals for the first time that the CCEM can indeed correctly identify the annotators' confusion characteristics and the desired ``ground-truth'' neural classifier under realistic conditions, e.g., when only incomplete annotator labeling and finite samples are available. Second, based on the insights learned from our analysis, two regularized variants of the CCEM are proposed. The regularization terms provably enhance the identifiability of the target model parameters in various more challenging cases. A series of synthetic and real data experiments are presented to showcase the effectiveness of our approach.