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>You must also explain why there isn’t a third point on the line, and why that line’s existence is guaranteed, which is not obvious (at least to me).

There isn't a third point on the line you found because the problem stipulates that the set of points is not collinear.

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So it says that not ALL points are on that single line. But it doesn't stipulate that THREE points can't be on the line---the theorem is about proving that you can find one that doesn't have the third point.

That's only assuming your set of points only contains three points. If you have four or more points, it's entirely possible that a line you draw between two points passes through a third.

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If you're given the above set of points, it's obviously not collinear due to the point at the top, but if you draw a line through any of the bottom two points, it will hit a third.

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