The tangent vector $\nu = \frac{d\lambda}{dt} \Bigr|_{\substack{p}}$ to a curve $\lambda(t) : R \to M$ at a point $p$ on a smooth manifold $M$ is the map from the set of real smooth functions $f$ defined in a neighborhood of $p$ to $R$, defined by
$\nu :f \to \frac{d(f \circ \lambda)}{dt} \Bigr|_{\substack{p}} $
My question is, why does one need the functions $f$?
Why not say the tangent vector is $\frac{d\lambda}{dt}\Bigr|_{\substack{p}}$ without reference to some functions on the manifold? What makes this argument naive/silly?
Is it because it simply cannot be defined this way? Why not use the coordinate system of the charts to map to $\mathbb{R}^n$ and use the homeomorphisms instead of $f$?
Also, sorry if this question is not a good one. I am a high school student, so I am self-learning this and I do not claim to be an expert. I know I barely know anything.