The unprecedented growth of machine-type devices has underscored the need for fundamental solutions to support emerging massive connectivity. In particular, unsourced random access (URA) has emerged as a promising paradigm, reframing the massive connectivity problem as a coding-theoretic challenge with favorable energy and spectral efficiency. Among the widely studied URA models, the Gaussian multiple-access channel (GMAC) and multi-input multi-output (MIMO) systems are of particular significance. Sparse code design is well-suited for URA, offering scalable solutions while retaining many advantages of legacy access protocols. However, existing sparse code designs often suffer from limited sparsity control, inefficient interference cancellation, and a strong dependence on specific channel code designs, posing challenges for long-term adaptability as more powerful channel codes continue to evolve. In MIMO-URA systems, additional activity detection and channel estimation phases typically lead to increased missed detection (MD) and false alarm (FA) errors compared with the GMAC model, which does not require these phases. While prior studies have predominantly focused on minimizing MD errors, the effective mitigation of FA errors remains an open problem. To address this challenge, we propose a sparse code with slotted transmission under the GMAC model, combined with an analytical power division strategy to enhance interference cancellation. Furthermore, we introduce a novel MIMO receiver framework based on joint pattern–data–channel (JPDC) estimation, which significantly reduces FA errors by leveraging the intrinsic correlation between user activity and transmitted data. Notably, the proposed method achieves improved overall system performance without requiring additional transmission overhead or complex algorithms.