Consider a manifold $M$ and a curve $\gamma : \mathbb{R} \supseteq I \rightarrow M$. The length of this curve is defined as
$$ L = \int \sqrt{g(X,X)}_{\gamma(t)} dt $$
where $g$ is the metric tensor, $X$ is the tangent vector to the curve and $t$ is the parameter of the curve.
I do not understand why we require a tangent space in order to calculate lengths of curves. I understand a metric tensor takes in two tangent vectors and spits out a number which allows us to calculate lengths and angles on manifolds, but why exactly do we require a metric defined in this way? Why is there no notion of length that is independent of tangent spaces and is in terms of the manifold alone?