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This paper investigates joint channel estimation and device activity detection in the LEO satellite-enabled grant-free random access systems with large differential delay and Doppler shift. In addition, the multiple-input multiple-output (MIMO) with orthogonal time-frequency space modulation (OTFS) is utilized to combat the dynamics of the terrestrial-satellite link. To simplify the computation process, we estimate the channel tensor in parallel along the delay dimension. Then, the deep learning and expectation-maximization approach are integrated into the generalized approximate message passing with cross-correlation--based Gaussian prior to capture the channel sparsity in the delay-Doppler-angle domain and learn the hyperparameters. Finally, active devices are detected by computing energy of the estimated channel. Simulation results demonstrate that the proposed algorithms outperform conventional methods.