Short answer: Planetary rotation establishes a preferential magnetic axis, and the field displays its instability in geological time scales, so it only seems stable.
A good description? Probably yes.
There's a wide variety (see this and this reviews) of magnetohydrodynamical models (with different assumptions, parameter ranges, boundary conditions, etc.) that successfully describe main characteristics of Earth's magnetic field dynamics. This fact suggests that only a small number of fundamental ingredients is needed to describe this dynamics, which puts it within reach of low-dimensional models. And, as the polarity reversals are strongly non-periodic and the flows expected to be turbulent, these models likely need to be chaotic.
That said, even if these models capture fundamental aspects of the relevant dynamics (they do fit the available measurements), they cannot fully reflect the complex flow underneath Earth's crust $-$ so they may be used to investigate major features of the dynamics, but probably not finer details.
why the Earth's magnetic field flips only after very long periods of time
Suppose we have a situation without flips, where either polarity is possible, but (completely) stable, i.e., there are no flips. In the lingo of chaos theory, that's described as the system having two attractors, which we can visualize as two separate blobs in phase space.
Now consider that, as a parameter is shifted, these two attractors can somehow merge, but only slightly, through a narrow bridge: a trajectory will move from one blob to the other with a frequency that's proportional to the effective width of this bridge. That's a way you can have long waiting times between the flips.
That's precisely what, e.g., Gissinger proposes in A new deterministic model for chaotic reversals (arxiv):
currently travelling about 64Km per year, but that seems small and also quite regular, not what you would expect to be the result of a chaotic system
As the picture above suggests, even if the times when the curve (trajectory) changes from one roll to the other are unpredictable, the overall movement, especially between changes is very smooth and approximately regular. It's a similar situation to the Solar system, which is chaotic, but whose short-time behavior is mostly pretty predictable.
why, instead of flipping, it does not simply change direction
Large scale flows in the planet are coupled to its rotation by the Coriolis effect, which "tends to organize the flow into rolls aligned along the north-south polar axis" (Wikipedia). Even if the flow is chaotic, the Coriolis effect is likely to remain a relevant influence.
Why is the intensity (as far as I know) also stable?
Actually "the intensity of the magnetic field decreases drastically during reversals" (Earth's Core and Lower Mantle, Jones et al, Chapter 4, Geomagnetic Reversals, C. Constable).