There is no local definition of a horizon. Horizons are everywhere, at least if you choose the right set of observers. One of those choices are e.g. Rindler coordinates corresponding to a particular set of uniformly accelerating observers in Minkowski space-time. In Schwarzschild coordinates you see a set of uniformly accelerating observers at $r=const.$ which would have to accelerate infinitely to stay at the horizon. Up to topology, you see the very same effect in Rindler coordinates. I.e. the significance of black-hole horizon follows, to a certain extent, from a globally topological aspect of the space-time.

There are of course various quasi-local definitions of a horizon which however 1) do not all give the same horizons in non-stationary space-times, and 2) do not always give the global "possible to escape to infinity" horizon in non-stationary space-times. In fact, if you have ever had a taste of the theory of dynamical systems, it would seem absurd that a rich dynamical theory allowing for chaos would unambiguously store the fate of the whole slice of the phase space (i.e. geodesics with all possible velocities at the given point) in one simple number.

The problem is that even the curvature invariants in fact correspond to certain "semi-local" quantitites, i.e. quantities not detectable on the tangent bundle $\sim \Delta x$ but in second-order differences $\sim \Delta x^2$. By this title, the Karlhede scalar corresponds to a "semi-semi-local" quantity measurable only through $\sim \Delta x^3$ effects.

This suggests that the scalar could be related to some of the quasi-local horizon definitions. I.e. the fact that the Karlhede scalar vanishes at the edge of the ergoregion in Kerr might be correlated with the fact that the ergoregion is the trapping surface for meridional-plane geodesics (which are transversal to the Killing vectors).

To understand curvature invariants, it is sometimes useful to introduce a 3+1 Bel decomposition with respect to some observers (I also recommend this paper by Cherubini et al. for a crash-course and application). In the case of the Karlhede scalar it would probably yield that it is some kind of combination of "slap-force gradients" and "local spatial-curvature variation" *with some kind of sign difference between the two*. In general, the zero of the Karlhede scalar will simply mean that the "slap force gradients" and "local spatial curvature variations" cancel each other out "by chance".

Of course, only direct investigation can reveal further insight.