Not all points being collinear does NOT mean that all 3-tuples of points are non-collinear! The hypothesis of the theorem is the former. And what it proves is that there is at least one such 3-tuple.

The other thread above helped me. You can have as many collinear points as you want as long as at least one point in the set is non-collinear.

Consider a 3x3 grid. It satisfies this argument.