Direct Device-to-Satellite (D2S) communications, which enable direct satellite connectivity with unmodified user equipment (UE), not only expand global coverage but also reshape the evolution of future access networks. However, D2S links face fundamental challenges due to inherently low signal-to-noise ratios (SNRs) and limited spatial multiplexing gains arising from near line-of-sight propagation, both of which severely constrain achievable spectral efficiency. Despite the lack of spatial multiplexing, this work shows that aggressive non-orthogonal transmissions, where multiple users (e.g., four) transmit concurrently over the same frequency resources, even to a single receive antenna, can unlock substantial capacity gains that remain entirely unexploited by existing systems. Realizing these gains in practice, however, requires receiver architectures that, to the best of our knowledge, have not yet been developed. To this end, we introduce NL-COMM-Sat, an efficient and flexible framework that overcomes this limitation by enabling aggressive non-orthogonal signal transmissions. In contrast to conventional non-orthogonal multiple access (NOMA) schemes, NL-COMM-Sat supports more than two UEs per receive antenna on the same frequency resource. The framework revisits optimal receiver design principles and proposes computationally efficient processing schemes that translate previously unexplored theoretical gains into tangible throughput improvements, even under realistic channel estimation errors and high-mobility Doppler conditions. Our evaluation shows that NL-COMM-Sat achieves up to a 2x increase in spectral efficiency compared to orthogonal multiple access and NOMA baselines across all considered SNR and Doppler regimes, even with a single-antenna receiver and user speeds of up to 500 km/h.