Understanding the relationships between the structure of polymer networks and their mechanical properties remains a long-standing challenge in polymer physics. In recent years, a new paradigm for network formation has emerged, whereby near-ideal hydrogels are produced by the cross-coupling of branched macromolecules with well-defined chain length. Such near-ideal networks constitute an excellent model system to revisit this question, as well as a promising platform for the design of new materials with tuneable properties. In this work, we systematically investigate the relative contributions of various network parameters (chain length, crosslink coordination, second-order loops) to the elasticity of near-ideal polymer networks using a computational random network model. Numerical results are compared to classical estimates of rubber elasticity theory. Our results highlight the role of the chain pre-stretch on the mechanical response, as well as the importance of topological defects on the elastic properties. We also compare our results to experimental data for near-ideal tetra-arm PEG hydrogels.