I absolutely despise these statements, that are perpetuated to the present day. I'm from near the Balkans, from Romania.
They depict the Balkan region as being some sort of backwards warmongers, which is not the case (yes, the region is not very developed but it's far from the poorest regions on this planet; also, yes, I know about the Yugoslav Wars, but that's a rather extreme scenario and even people in the region were appalled at what the Serbs did, especially).
Anyway, the reason I don't like it: it was foreign empires vying in the region that made the mess. Turks, Austrians, Russians, Brits, Italians, Germans, French, they're the ones that started the wars. There was no way the local conflicts would have ever become global conflicts. The locals just didn't have the desire or resources to escalate such a conflict. Instead these empires just used them as battering rams against the other empires.
The statement is true and you give yourself a very good description of it. He did not say "it will come of some damned fools" but "some damned foolish thing in the Balkans", i.e. the power play between Russia and Austria-Hungary.
In my eyes the main difference between those and the Balkans was that the former were mostly allowed to blow themselves up. Peace only came when they got tired of it. In the Balkans they got tired of it in 1999. They couldn't do it earlier because of... empires :)
Of course, there are no guarantees in life, but if I were a betting man I'd say that for each extra peaceful year of peace in the Balkans you can add 1 year of peace in the future.
So my forecast is for at least 19 years of peace :)
The reality is simpler, there was a group of people in Austria who wanted war with Serbia, and a group of people in Germany who wanted war with Franz and Russia. Unfortunately for world history those people happen to be in particular places at that time, and so they started the war.
It is really that simple, a number of people wanted war, so they started it.
It is, of course, possible that even Bismarck could not have continued to manage them given the strains of nationalism that he set in motion. However, Bismarck was very motivated to maintain the peace rather than motivated to engage in war and that served Germany very well for a remarkably long time.
Even worse, Bismarck understood about not letting things get out of hand. He was quoted in 1898--"One day the great European War will come out of some damned foolish thing in the Balkans"