The Sylvester–Gallai Theorem

I think what's going on here is that you've misunderstood the theorem's hypothesis. The hypothesis isn't that no three of the points are collinear; rather, it's the weaker statement that there isn't any one single line that all the points lie on. It's true that with your version of the hypothesis the theorem would be trivial; but with the actual hypothesis it is is nontrivial.

Try to come up with a set non-colinear points where NO line passes through two and ONLY TWO points and you'll see the value of the statement.

You may think "I'm sure I can arrange these points in a way where EVERY line will cross three or more points" but you will fail if you try unless ALL points are colinear.

It’s that the line passes through exactly two points, which if you think about it is not exactly obvious.

> So what is the new fact?

For all arbitrarily sized (but finite) sets of not collinear points, there's always a line that passes through exactly two points in the set.

It can help to think about theorems like this by restating them as a puzzle asking for a counterexample.

Given N points, N > 2, can you arrange them in a Euclidean plane so that (1) they are not all on the same line, and (2) every line that goes through two of the points must also go through at least one more of the points?

The theorem says that you cannot do this.