Living cells exhibit nonequilibrium dynamics emergent from the intricate interplay between molecular motor activity and the viscoelastic cytoskeletal matrix. The deviation from thermal equilibrium can be quantified through frequency-dependent effective temperature or time-reversal symmetry breaking quantified, e.g., through the Kullback-Leibler divergence. Here, we investigate the fluctuations of an AFM tip embedded within the active cortex of mitotic human cells with and without perturbations that reduce cortex activity through inhibition of material turnover or motor proteins. While inhibition of motor activity significantly reduces both effective temperature and time irreversibility, inhibited material turnover leaves the effective temperature largely unchanged but lowers the time irreversibility and entropy production rate associated with the fluctuation-dissipation theorem violation of tip dynamics. Our experimental findings in combination with a minimal model highlight that time irreversibility, effective temperature, and entropy production rate can follow opposite trends in active living systems, challenging, in particular, the validity of effective temperature as a proxy for the distance from thermal equilibrium, particularly in the presence of mechanical changes. Furthermore, we propose that biological activity regulates the occurrence of time-asymmetric deflection spikes in the dynamics of observables, providing a previously unrecognized link between entropy production and time irreversibility.