This is related to Mori's theorem through 
Grauert's ampleness criterion in 
Hartshorne's "Ample vector bundles" (Proposition 3.5).
Let's assume that $M$ is projective and $\dim M  \ge 2$. 
Let $\alpha : T^*M \to Y$ denote the affinization of $T^*M$.
To show that $TM$ is ample, according to the criterion
it suffices to show that 
the zero section $M \subset T^*M$ is the only subvariety
contracted by $\alpha$.

Assume to the contrary that $\alpha$ contains a fiber $F$ such that $F \ne M$.
Since $\alpha$ is a symplectic resolution and $\dim M  \ge 2$, 
it is a small contraction, so 
the image of the exceptional locus of $\alpha$ in $Y$ is the singular locus of $Y$.
Now as $F \ne M$ and the intersection of $F$ with each fiber of $T^*M \to M$
is finite, $F$ is not stable under the $\mathbf{C}^*$-action on $T^*M$.
Since the $\mathbf{C}^*$-action descends to $Y$, 
it implies that the singular locus of $Y$ contains $\alpha(\mathbf{C}^* \cdot F)$,
which contradicts the assumption that $Y$ has only isolated singularities.