To exploit the benefits of massive multiple-input multiple-output (M-MIMO) technology in scenarios where base stations (BSs) need to be cheap and equipped with simple hardware, the computational complexity of classical signal processing schemes for spatial multiplexing of users shall be reduced. This calls for suboptimal designs that perform well the combining/precoding steps and simultaneously achieve low computational complexities. An approach based on the iterative Kaczmarz algorithm (KA) has been recently investigated, assuring well execution without the knowledge of second order moments of the wireless channels in the BS, and with easiness, since no tuning parameters, besides the number of iterations, are required. In fact, the randomized version of KA (rKA) has been used in this context due to global convergence properties. Herein, modifications are proposed on this first rKA-based attempt, aiming to improve its performance-complexity trade-off solution for M-MIMO systems. We observe that long-term channel effects degrade the rate of convergence of the rKA-based schemes. This issue is then tackled herein by means of a hybrid rKA initialization proposal that lands within the region of convexity of the algorithm and assures fairness to the communication system. The effectiveness of our proposal is illustrated through numerical results which bring more realistic system conditions in terms of channel estimation and spatial correlation than those used so far. We also characterize the computational complexity of the proposed rKA scheme, deriving upper bounds for the number of iterations. A case study focused on a dense urban application scenario is used to gather new insights on the feasibility of the proposed scheme to cope with the inserted BS constraints.