This paper investigates the rightmost eigenvalue behavior of a class of single-delay large-scale consensus dynamics where information shared among the agents is delayed. We present an analytical approach that links stabilizing effects of delays to the finite number of graph Laplacian eigenvalues associated with the network and to the coupling strengths between the agents. In particular, we extend our previously developed concept called Responsible Eigenvalue (RE), which is concerned with the delay margin of the dynamics, to the analysis of how the infinitely many eigenvalues of the dynamics are configured on the complex plane. We show that, under certain conditions of the graph Laplacian eigenvalues, the rightmost eigenvalue can be placed farther from the imaginary axis of the complex plane for larger delays. This information leads to rules by which coupling strengths of the agents can be designed to reduce the settling time of the system, rendering the consensus dynamics delay tolerant. Numerical examples are provided to demonstrate the stabilizing effects of delays, including uncertainties in delays, and simulation results are presented to show the improvements in transient response of the consensus dynamics.