Cells use their dynamic actin network to control their mechanics and motility. These networks are made of branched actin filaments generated by the Arp2/3 complex. Here we study under which conditions the microscopic organization of branched actin networks builds up a sufficient stress to trigger sustained motility. In our experimental setup, dynamic actin networks or "gels" are grown on a hard bead in a controlled minimal protein system containing actin monomers, profilin, the Arp2/3 complex and capping protein. We vary protein concentrations and follow experimentally and through simulations the shape and mechanical properties of the actin gel growing around beads. Actin gel morphology is controlled by elementary steps including "primer" contact, growth of the network, entanglement, mechanical interaction and force production. We show that varying the biochemical orchestration of these steps can lead to the loss of network cohesion and the lack of effective force production. We propose a predictive phase diagram of actin gel fate as a function of protein concentrations. This work unveils how, in growing actin networks, a tight biochemical and physical coupling smoothens initial primer-caused heterogeneities and governs force buildup and cell motility.