Dear Undergraduate Student, first of all, congratulations on the beautiful geometry you chose to study so early in your studies.

To prove $dim X_3 \leq 2$ it is indeed enough to prove that there exists a secant to $C$ which is not a trisecant, since the space of secants is irreducible (Harris, Algebraic Geometry, p.144).

This existence is proved on page 110 of the book Geometry of Algebraic Curves by Arbarello, Cornalba, Griffiths, Harris (Springer Grundlehren 267).They assume the curve is nondegenerate i.e. not contained in a hyperplane, and so certainly not contained in a plane. For a plane curve the result $dim X_3 \leq 2$ is evident.

Dear Undergraduate Student, first of all, congratulations on the beautiful geometry you chose to study so early in your studies.

To prove $dim X_3 \leq 2$ it is indeed enough to prove that there exists a secant to $C$ which is not a trisecant, since the space of secants is irreducible (Harris, Algebraic Geometry, p.144).

This existence is proved on page 110 of the book Geometry of Algebraic Curves by Arbarello, Cornalba, Griffiths, Harris (Springer, Grundlehren 267).They assume the curve is nondegenerate i.e. not contained in a hyperplane of $\mathbb P^n, n \geq 3$, and you can reduce to this for a non plane curve. For a plane curve the result $dim X_3 \leq 2$ is evident.