Killing fields on product metrics

Use the Corollary of this paper:

If a compact Riemannian manifold $M$ splits as $M = M_1 \times M_2$, then the identity component of the isometry group splits as $I_0 (M) = I_0(M_1) \times I_0(M_2)$.

(This is authors' stated Corollary, but from their main theorem it looks like we can weaken compactness to your requirement that $M$ has no Euclidean factor.)

Let $\zeta_t : \mathbb R \to I_0(M)$ be the one-parameter group of isometries generated by $Z$. The above fact means we can write $\zeta_t = (\xi_t, \upsilon_t)$ with $\xi_t$ acting on $M_1$, $\upsilon_t$ acting on $M_2$. Since $\zeta_t$ is a one-parameter subgroup, both $\xi_t$ and $\upsilon_t$ must be as well, and their generators $X,Y$ will satisfy $X+Y=Z$.