Interfacial swimmers are objects that self-propel at an interface by autonomously generating a gradient of surface tension, often through the continuous release of a surfactant. While the case of asymmetric swimmers has long been studied, experiments have shown that spontaneous motion is also possible for symmetric swimmers. The basic mechanism of symmetry-breaking is qualitatively well-established but one key aspect of the phenomenon that has proved particularly difficult to elucidate is the role of Marangoni effects in the self-propulsion. We address this question by numerical methods, which can fully handle the complex interplay between swimmer motion, fluid flow, surfactant distribution, and Marangoni stresses. Our swimmer is a disk releasing a soluble surfactant in a deep-layer fluid. We investigate how the swimming velocity, represented by a Péclet number Pe * depends on its characteristics, as encapsulated in the Marangoni number M. We analyze the properties of the swimming diagram Pe * (M) and compare with approximate models to understand their origin. We find that the low-Pe * regime exhibits a bistability region: spontaneous swimming involves a threshold Marangoni number, a discontinuity in velocity and possibly hysteresis. Those features are present only for a full description of the problem and reveal the subtle but key role of Marangoni flows. The large-Pe * regime features a robust asymptotic scaling law Pe * ∼ M α , whose exponent α 0.72 is close to the 3/4 value predicted by a simplified model, indicating a much weaker influence of Marangoni flows. While our results were obtained assuming a point-source swimmer in the Stokes flow regime, we show that the picture remains very similar when considering a spatially extended source size, finite Reynolds number, or a fixed concentration swimmer. We discuss our findings in relation to experiments.