Graphynes can be structurally envisioned as 2D extensions to graphene, whereby linearly bonded carbon linkages increase the distance between trigonal carbon nodes. Many graphynes have been predicted to exhibit a Dirac-like semimetallic (SEM) graphenic electronic structure, which could potentially make them competitive with graphene for applications. Currently, most graphynes remain as attractive synthetic targets, and their properties are still unconfirmed. Here, we demonstrate that the electronic structure of hexagonal α-graphyne is analogous to that of biaxially strained graphene. By comparison with accurate quantum Monte Carlo results on strained graphene, we show that the relative energetic stability of electronic states in this correlated 2D system can be captured by density functional theory (DFT) calculations using carefully tailored hybrid functionals. Our tuned hybrid DFT approach confirms that α-graphyne has a low energy correlated Mott-like antiferromagnetic insulating (AFI) state, which competes with the SEM state. Our work shows that the AFI-SEM crossover in α-graphyne could be tunable by in-plane biaxial strain. Applying our approach to other graphynes shows that they should also exhibit correlated AFI states, which could be dominant even at zero strain. Calculations using an onsite Coulombic repulsive term (i.e., DFT + U) also confirm the predictions of our hybrid DFT calculations. Overall, our work strongly suggests that graphynes are not as graphenic (i.e., Dirac-like) as often previously predicted by DFT calculations using standard generalized gradient approximation functionals. However, due to the greater electronic versatility (e.g., tunable semiconducting bandgaps and accessible spin polarized states) implied by our study, graphynes could have novel device applications that are complementary to those of graphene.