Varieties with an ample vector bundle mapping to their tangent bundle

This is not a complete answer, but it's way too long for a comment.

You can define ampleness for arbitrary coherent sheaves. This is done for example in V. Ancona, "Faisceaux amples sur les espaces analytiques". The definition is the same as in the locally free case: A coherent sheaf $\mathscr E$ on a compact complex space $X$ (say, nonzero and torsion-free) is said to be ample if for any coherent sheaf $\mathscr F$ on $X$ there exists $n_0$ such that for $n \geq n_0$ the sheaf $\mathscr F \otimes S^n \mathscr E$ is globally generated.

This is equivalent to $\mathscr F \otimes S^n \mathscr E$ having vanishing higher cohomology groups for $n \gg 0$, and to the line bundle $\mathscr O_{\mathbb P(\mathscr E)}(1)$ on $\mathbb P_X(\mathscr E)$ being ample (Proposition 2.5 in Ancona's paper).

Now a nonzero quotient of an ample vector bundle will be ample in this general sense. Hence your question becomes: if $T_X$ contains an ample subsheaf $\mathscr F$, then is $X$ isomorphic to a projective space? In case the Picard number $\rho(X) = 1$, this is Corollary 4.3 in "Galois coverings and endomorphisms of projective varieties" by Aprodu, Kebekus, and Peternell. In fact they show that in this case, $\mathscr F$ is already locally free and then they apply the result of Andreatta and Wiśniewski.

The answer to this question is now known to be yes -- it is Corollary 1.2 of this paper of Jie Liu.