>> Adding points to the set can only ever shrink the size of the smallest triangle that the set defines.

> I don’t think this is true. Any 3 points almost in a line will have a very small area. Adding points inside the triangle would help

Come on. Adding points can never help. This is not a difficult proof:

Let A be a set of points, and let T be the set of all triangles whose vertices are drawn from A.

Let A' be any set of points that is a superset of A.

Let T' be the set of all triangles whose vertices are drawn from A'.

We can immediately observe that, since A is a subset of A', T is likewise a subset of T'.

Therefore, the smallest triangle in T' can never be larger than the smallest triangle in T, because the smallest triangle in T is a member of T'.

> I can think of a few other examples.

You can claim to, but you won't be telling the truth. This is a simple case of the obvious principle that if you're playing against an omniscient opponent, giving more options to them can't help you.