Killing vectors in Schwarzschild's metric

In this particular metric: $$ds^2=\frac{1}{z}(-dt^2 +dx^2+dy^2)+zdz^2$$ we have the three Killings $$\xi_1=\partial_t$$ $$\xi_2=\partial_x$$ $$\xi_3=\partial_y$$and we also have $$\xi_4=x \partial_y-y \partial_x$$ $$\xi_5=x\partial_t-t\partial_x$$ $$\xi_6=y\partial_t-t\partial_y$$

However, in the Schwarzschild metric, we of course have $$\xi_1=\partial_t$$ $$\xi_2=\partial_\phi$$ Why don't we have $$\xi_3=t \partial_\phi-\phi \partial_t$$

Your metric contains the piece $-\mathrm dt^2 +\mathrm dx^2+\mathrm dy^2$ which is invariant under rotations in the $(t,x,y)$ plane. Thus, the generators of these rotations are conserved.
The Schwarzschild metric, $$\mathrm ds^2\sim f(r)\mathrm dt^2+g(r,\theta)\mathrm d\phi^2+\cdots$$ is not invariant in the $(t,\phi)$ plane. Thus, there is no conservation law associated to these rotations, and no Killing field.