The β-relaxation dynamics in a simple monatomic Lennard-Jones system is re-visited for both quenching and crushing by the molecular dynamics technique. We obtain for each process the pairwise liquid structures and use them as input data to numerically solve for the dynamical transition point within the idealized version of the mode-coupling theory. It is found that near the dynamical transition point the dynamical behavior of the Lennard-Jones liquid such as the master function, the tagged-particle distribution function, etc. deviates more from a purely repulsive hard-sphere system but is closer to that of a liquid metal. This indicates, from the details of inter-particle interactions that, the attractive tail of the pair potential does play a non-negligible role in the feedback effects addressed in the mode-coupling theory. To quantify our studies, we have performed simulation also for the self-part intermediate scattering function for a temperature range that spans over the dynamical transition point. Compared with the prediction of mode-coupling theory, the simulated tagged particle self-part density-density correlation function displays a sluggish and stretched relaxation, although its retarded signature is observed to be extremely weak. This implies that a simple monatomic liquid because of this basic nature of interparticle interactions is structurally much more difficult to exemplify the β-relaxation process.