The economic essence of saddle-path stability, is that it requires conscious behavior and decision making from the part of economic agents, in order for the system to remain on the saddle path. Simply put, we cannot do "whatever we like" and still expect the system to settle down. But that is what makes it interesting (and true to reality): on their own, economic systems are unstable.
Contrast this with the basic Solow growth model: there, fixing arbitrarily the savings rate (i.e. depriving ourselves from making a decision on it period per period) makes the system properly stable. But once we want to optimize with respect to the savings rate, we find out that there is a single path that leads to long-term equilibrium.
Now, perusing the pages of a book like Barro and Sala-i-Martin's "Economic Growth", which contains many-many extensions of the basic RCK growth model, one will not find a model with intertemporally optimizing agents that is also mathematically stable.
Elaborating, in response to a thoughtful comment by the OP, my approach to the issue of "stability" of an economy is the following: The fact that economies tend to "correct course" is not an indication, let alone proof, that they posses some inherent stabilization mechanism independent of the purposeful action of the economic agents.
If this was the case, then indeed, we should have models that possessed the mathematical property of stability (I remind again that "saddle-path stable" although it uses the word "stable", is a mathematically unstable system).
On the other hand, economies do not plunge into chaos every day. The solution to the puzzle from my point of view is not that we "continuously fine tune the economy" -such dreams and visions have been tried and squashed (irrespective of what ideological/philosophical (dis) agreements one may have with socialism, the factual lesson to be learned from the failure of centrally planned economies is that an economy is such a complex system that no one can "plan and then fine-tune it", especially in our era).
My answer is that economies, after being hit by external shocks or internal institutional failures or technology upheavals, or.., tend to return to a dynamic path that tends to some fixed point by the aggregate consequences of decentralized purposeful action of economic agents that try not to fix the economy as a whole, but their own economic present and future. Economies survive not "because they inherently do", not because the government is "steering the boat with a certain and effective hand", but because individuals try hard every day to survive.
And the RCK models reflect just that: if one performs comparative statics on it, assuming an external shock that suddenly puts the economy out of the saddle-path, the prediction of the model as to what will happen in order for the economy to return to the (new) saddle path towards the (new) long-run equilibrium, is more or less what we observe in reality. But this happens not because the government makes it so, but because individual economic agents realize that they must do so in order not to eventually self-destruct.
I will indulge myself and I will link to a little exposition I crafted a year ago for a class of graduate students, to show them how the simplest RCK model can help them think about the economic crisis (and subsequent depression) in Greece -which was sudden enough to be described as a shock. The exposition can be found here.