Time delay and disturbances are commonly encountered in real-world application systems, and their existence significantly hampers the operation of various controllers available in the discipline of control systems theory. Time-delay estimation and disturbance compensation are closely related to obtaining the desired system stability and efficient controller operation. This paper discusses state-of-the-art methods for time-delay estimation (TDE)-based methods related to active disturbance rejection control (ADRC) methods and proportional-integral-derivative (PID) controllers found in the latest literature on control systems theory. The methodology includes simulation designs incorporating the integration of predictive extended state observer-based ADRC (PESO-ADRC) and conventional PID controllers with the TDE mechanism, followed by their respective control on systems with time delay and disturbances. A comparative analysis performed on the TDE compensation methods highlights that TDE enhances the robustness to time delay, various uncertainties, and nonlinear dynamics existing in the controlled system. The time-varying delay, nonlinear backlash-like hysteresis, and an added system external disturbance were considered in the simulation. The performance was measured based on specific transient response characteristics such as the rise time, settling time, overshoot criteria, and performance index measures such as the integral of time-weighted absolute error (ITAE) and the percentage of improvement by the decrease in overshoot given by $P_i$ (%). Further, a sensitivity analysis of TDE parameters to the controllers' operation was also performed. Experimental results indicate stability and a strong capacity to regulate the transient and steady-state responses under the impact of various uncertainties. Therefore, the comparative analysis conducted between TDE-ADRC and TDE-PID control methods signifies the importance of TDE with disturbance compensation in time-delayed systems, commonly found in real-world industrial applications.