We investigate the competition between glass formation and crystallization of open tetrahedral structures for particles with tetrahedral patchy interactions. We analyze the outcome of such competition as a function of the potential parameters. Specifically, we focus on the separate roles played by the interaction range and the angular width of the patches, and show that open crystal structures (cubic and hexagonal diamond and their stacking hybrids) spontaneously form when the angular width is smaller than about 30°. Evaluating the temperature and density dependence of the chemical potential of the fluid and of the crystal phases, we find that adjusting the patch width affects the fluid and crystal in different ways. As a result of the different scaling, the driving force for spontaneous self-assembly rapidly grows as the fluid is undercooled for small-width patches, while it only grows slowly for large-width patches, in which case crystallization is pre-empted by dynamic arrest into a network glass.