The emergence of new genes is an important driver of evolutionary novelty. Yet, we lack a conceptual and computational approach that accurately traces gene-family founder events and effectively associates them with trait innovation and major radiation events. Here, we present GenEra, a DIAMOND-fuelled gene-family founder inference framework that addresses previously raised limitations and biases of founder gene detection in genomic phylostratigraphy by accounting for homology detection failure (HDF). We demonstrate how GenEra can accelerate gene-family founder computations from several months to a few days for any query genome of interest. We analyzed 30 genomes to explore the emergence of new gene families during the major evolutionary transitions in plants, animals and fungi. The detection of highly conserved protein domains in these gene families indicates that neofunctionalization of preexisting protein domains is a richer source of gene-family founder events compared with de novo gene birth. We report vastly different patterns of gene-family founder events in animal and fungi before and after accounting for HDF. Only plants exhibit a consistent pattern of founder gene emergence after accounting for HDF, suggesting they are more likely to evolve novelty through the emergence of new genes compared to opisthokonts. Finally, we show that the transition to multicellularity in streptophytes, the terrestrialization of land plants and the origin of angiosperms are associated with gene-family founder bursts, as well as the evolution of bilateral symmetry in animals.