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We use the Schwarzschild radius $r_s$ (rather than a diameter) because it's convenient. We're want to describe what happens in the vicinity of a black hole, so it's natural to talk about the distance from the black hole. For example, there's the photon sphere at $3r_s/2$ (for a Schwarzschild BH), and the innermost stable circular orbit or ISCO, the smallest ...
The diameter of a black hole (the distance "through" it) or the radius (distance to the center) aren't really meaningful concepts. The singularity is in the future, not in any spatial direction. The $r$ Schwarzschild coordinate is actually a so-called "reduced circumference" – the metric length of a circle around the hole, divided by $2π$ ...