In my answer to https://physics.stackexchange.com/questions/225436/why-are-killing-fields-relevant-in-physics I wrote:
The flow of a Killing field can drag the whole manifold. Since Killing fields are divergence free and thus their flows have no sources or sinks, dragging the whole manifold gives a bijection between the start and end points (an automorphism). Since the flow preserves the metric the bijection is an isometry.
Which was convincing to me at the time and also sounded pretty standard from what I have on read on the subject. However, after writing it I have thought of what appears to be a trivial counter example.
Consider a constant vector field $v$ on the semi-infinite open interval $(0,\infty)$. Clearly $v$ is a Killing field and (trivially) also divergence free. The flow of $v$ has no sources or sinks. It is an isometry since the definition of isometry only requires injectivity. But the flow does not generate symmetries, which need to be isomorphisms. If the field is pointing towards $\infty$ then the isometries are not onto, since nothing maps into $(0, vt]$ and if it points towards $0$ the isometry is not even defined since the points in $(0, vt]$ fall of the manifold.
It seems like I missed something really basic in my understanding of the subject somewhere, but what?
Did I miss a compactness requirement somewhere?
One way or another, you need the flow $\varphi_t$ to be defined in some open interval around $0$, which isn't the case in your example. Then for sufficiently small $\varepsilon$, we'll have that $\varphi_{\epsilon}$ is an isometry with inverse $\varphi_{-\epsilon}$.