The strategic choice of model "openness" has become a defining issue for the foundation model (FM) ecosystem. While this choice is intensely debated, its underlying economic drivers remain underexplored. We construct a two-period game-theoretic model to analyze how openness shapes competition in an AI value chain, featuring an incumbent developer, a downstream deployer, and an entrant developer. Openness exerts a dual effect: it amplifies knowledge spillovers to the entrant, but it also enhances the incumbent's advantage through a "data flywheel effect," whereby greater user engagement today further lowers the deployer's future fine-tuning cost. Our analysis reveals that the incumbent's optimal first-period openness is surprisingly non-monotonic in the strength of the data flywheel effect. When the data flywheel effect is either weak or very strong, the incumbent prefers a higher level of openness; however, for an intermediate range, it strategically restricts openness to impair the entrant's learning. This dynamic gives rise to an "openness trap," a critical policy paradox where transparency mandates can backfire by removing firms' strategic flexibility, reducing investment, and lowering welfare. We extend the model to show that other common interventions can be similarly ineffective. Vertical integration, for instance, only benefits the ecosystem when the data flywheel effect is strong enough to overcome the loss of a potentially more efficient competitor. Likewise, government subsidies intended to spur adoption can be captured entirely by the incumbent through strategic price and openness adjustments, leaving the rest of the value chain worse off. By modeling the developer's strategic response to competitive and regulatory pressures, we provide a robust framework for analyzing competition and designing effective policy in the complex and rapidly evolving FM ecosystem.