Over the past two decades, comparative sequence analysis using codon-substitution models has been honed into a powerful and popular approach for detecting signatures of natural selection from molecular data. A substantial body of work has focused on developing a class of "branch-site" models which permit selective pressures on sequences, quantified by the ω ratio, to vary among both codon sites and individual branches in the phylogeny. We develop and present a method in this class, adaptive branch-site random effects likelihood (aBSREL), whose key innovation is variable parametric complexity chosen with an information theoretic criterion. By applying models of different complexity to different branches in the phylogeny, aBSREL delivers statistical performance matching or exceeding best-in-class existing approaches, while running an order of magnitude faster. Based on simulated data analysis, we offer guidelines for what extent and strength of diversifying positive selection can be detected reliably and suggest that there is a natural limit on the optimal parametric complexity for "branch-site" models. An aBSREL analysis of 8,893 Euteleostomes gene alignments demonstrates that over 80% of branches in typical gene phylogenies can be adequately modeled with a single ω ratio model, that is, current models are unnecessarily complicated. However, there are a relatively small number of key branches, whose identities are derived from the data using a model selection procedure, for which it is essential to accurately model evolutionary complexity.