Advances in machine learning have led to the development of foundation models for atomistic materials chemistry, enabling quantum-accurate descriptions of interatomic forces across diverse compounds at reduced computational cost. Hitherto, these models have been benchmarked relying on descriptors based on atoms' interaction energies or harmonic vibrations; their accuracy and efficiency in predicting observable and technologically relevant heat-conduction properties remains unknown. Here, we introduce a framework that leverages foundation models and the Wigner formulation of heat transport to overcome the major bottlenecks of current methods for designing heat-management materials: high cost, limited transferability, or lack of physics awareness. We present the standards needed to achieve first-principles accuracy in conductivity predictions through model's fine-tuning, discussing benchmark metrics and precision/cost trade-offs. We apply our framework to a database of solids with diverse compositions and structures, demonstrating its potential to discover materials for next-gen technologies ranging from thermal insulation to neuromorphic computing.