Trade competitiveness can coexist with structurally fragile value chains. When chain feasibility fractures from trade competitiveness, competitiveness without coherence becomes sustainability’s opposite. This paper proposes revisiting the concept of sustainability in agri-food systems, through the lens of structural coherence, understood as the alignment between trade competitiveness, export-destination diversification, and value chain capacity. The research goal is to design and operationalize a diagnostic instrument for structural coherence testing through the triangulation of constant market share analysis (CMSA), the Herfindahl–Hirschman Index (HHI), and physical structural input–output analysis (I-OA). CMSA measures two elements: demand- and competitiveness-driven export dynamics. Export patterns are further explored to verify if there are any destination-market concentration risks (HHI). I-OA closes the loop by linking trade outcomes to internal value chain capacity and efficiency. With clear upstream–downstream segmentation, the sunflower oilseed value chain of the European Union (EU) represents an empirically fertile ground, relevant in the context of the geopolitical disruptions of Black Sea trade corridors and double-cropping dynamics with food-fuel and land-use trade-offs. Focusing on Bulgaria, France, Hungary, Romania, and Spain, which collectively account for more than 85% of EU sunflower seed production, this paper benchmarks post-2013 Common Agricultural Policy (CAP) programming effects, utilized as a proxy for a period of stability, against the post-2020 window, marked by a sequence of crises. Diagnosis is facilitated through findings triangulation, enabling deriving CAP-relevant policy recommendations, aligned with country-specific binding constraints. Results show heterogeneous structurally incoherent profiles: Bulgaria suffers from growth-induced stress, France’s chain efficiency is eroded, the Hungarian chain lacks competitiveness, Romania is raw-export dependent with value-added leakage, and Spain is structurally constrained by physical limits. Policy recommendations target reorienting market-driven low value-added trade behaviors toward structurally sustainable value chain trajectories.