How do cations control the crystal structure of silicate minerals?
I would say that cations would control the structure in several ways; in the case of aluminum octahedra in aluminosilicates, for example, they might completely influence it by causing the presence of completely separate alternating layers of bonded octahedra. The octahedra may even dominate the unit-cell, appearing in a larger numerical and volumetric capacity than the tetrahedra.

Additionally, the different bonded cations have different polarities, and by extension, exert different forces on the relevant bonds; this plays a huge role in determining how the structure responds under conditions of high pressure relevant to the Earth's interior, e.g determining where shearing takes place, etc.