Is there a Lagrangian action that leads to gradient descent?

Let us rewrite OP's eq. (3) as $$ \dot{x}^j~=~-g^{jk}(x)\frac{\partial V}{\partial x^k} \qquad \Leftrightarrow \qquad g_{jk}(x)\dot{x}^k~=~-\frac{\partial V}{\partial x^j}. \tag{3}$$ We note that this dynamics has no time-reversal symmetry.

Moreover, eq. (3) implies that $$ V(x_i)-V(x_f) ~=~ \int_{t_i}^{t_f} \! \mathrm{d}t ~\dot{x}^jg_{jk}(x)\dot{x}^k~\geq~0 . $$ We conclude that this dynamics cannot have closed orbits, and that it is dissipative in nature.

If we consider a sufficiently small neighborhood, we can use Riemann normal coordinates, and assume that the metric components $g_{jk}$ are constant. We can even assume that the metric components $g_{jk}$ are diagonal. By furthermore scaling the $x^j$-coordinates, we may assume that the metric components $g_{jk}=\delta_{jk}$. More generally, this is known as Aristotelian mechanics, which has no conventional stationary action principle. See also this Phys.SE post.