Quite clearly, the answer to the question
Is the assertion that "a Lorentzian manifold in consideration does not admit a global, non-vanishing, asymptotically time-like Killing vector field" a coordinate-independent statement?
Is "most certainly". This can easily be seen from the fact that the statement does not involve any coordinate dependent concepts.
If yes, what goes wrong in the example shown above?
This is harder to explain. (Partially because there seem to be several misconceptions).
First I'll try to explain what is actually going on. De Sitter space is a maximally symmetric spacetime. Consequently, there exists a complete set of four independent Killing vector fields. Or more precisely there exists a four dimensional family of Killing vector fields. Consequently, at any point (event) in deSitter space, you will be able to find a Killing vector field that is timelike at that event.
However, that Killing vector field will not be timelike everywhere but only on some local patch of de Sitter space. Each of these patches corresponds to a patch of static coordinates (which uses the integral curves of that particular Killing vector field as its "time" coordinate). The edge of this region is the particle/event horizon for some observer. The location of these horizons is clearly observer dependent. 298
This does however not mean that the dynamics of spacetime is observer dependent.
This brings us to the central misconception in you example. You write:
According to <PERSON>, the region $r\gt H_\Lambda^{-1}$ is described by the following metric:
$$ ds^2 = \frac{dt^2}{(H_\Lambda t)^2 - 1} -\Big((H_\Lambda t)^2 -1\Big) dr^2 -t^2 d\Omega^2, \qquad t\gt H_\Lambda^{-1} \,. $$
Here, you have extended the static coordinate metric through its coordinate singularity(ies) at the horizon(s). This is obviously not how to do this. To analytically extend <PERSON>'s metric, he would need to extend it through the horizons at $t=\pm\infty$. That is, he would need to change to coordinates that are regular at the horizon. E.g. the "closed slicing" coordinates. When he does this he finds that he expects the spacetime behind the horizon to be perfectly similar to the spacetime at his local patch.
Another note. Somewhere in the comments you write (as an alternative formulation of your question):
Given a Lorentzian manifold (with signature +---) and a chart such that the metric is (locally expressed) a function of time, can we always find another chart with the same domain such that the metric is time-independent?
No this is not always possible. Doing this requires the existence of a Killing vector field that is timelike, at least locally in the patch where you want to make the metric time-independent. 976