Integrated sensing and communications (ISAC) is one of the key enabling technologies in future sixth-generation (6G) networks. Current ISAC systems predominantly rely on deterministic pilot signals within the signal frame to accomplish sensing tasks. However, these pilot signals typically occupy only a small portion, e.g., 0.15% to 25%, of the time-frequency resources. To enhance the system utility, a promising solution is to repurpose the extensive random data payload signals for sensing tasks. In this paper, we analyze the ISAC performance of a multi-antenna system where both deterministic pilot and random data symbols are employed for sensing tasks. By capitalizing on random matrix theory (RMT), we first derive a semi-closed-form asymptotic expression of the ergodic linear minimum mean square error (ELMMSE). Then, we formulate an ISAC precoding optimization problem to minimize the ELMMSE, which is solved via a specifically tailored successive convex approximation (SAC) algorithm. To provide system insights, we further derive a closed-form expression for the asymptotic ELMMSE at high signal-to-noise ratios (SNRs). Our analysis reveals that, compared with conventional sensing implemented by deterministic signals, the sensing performance degradation induced by random signals is critically determined by the ratio of the transmit antenna size to the data symbol length. Based on this result, the ISAC precoding optimization problem at high SNRs is transformed into a convex optimization problem that can be efficiently solved. Simulation results validate the accuracy of the derived asymptotic expressions of ELMMSE and the performance of the proposed precoding schemes. Particularly, by leveraging data payload signals for sensing tasks, the sensing error is reduced by up to 5.6 dB compared to conventional pilot-based sensing.