I suspect dsalerno5 is right... President Wilson, while admirable in his desire to make the world safe for democracy, as delineated in his Fourteen Points, was ultimately proved naive. He was double crossed by his colony-hungry allies in Britain and France and by his own Congress. The Allies refused to negotiate with the Kaiser and insisted upon self determination among the peoples of the Austro-Hungarian Empire, which was fracturing anyhow. With both of those emperors gone, a huge power vacuum was created, into which Hitler was able to waltz with relative ease, being democratically elected. The German people were not prepared to live in a democratic republic. That had never been their tradition. An example of that is Russia today under Putin... he has steadily chipped away at Russian "democracy" because that is a system the Russian people are unfamiliar with. Thus, they are devolving back into a popular and familiar authoritarianism. I think the Allies learned their lesson after World War I when at the end of World War II, the Japanese were allowed to retain their monarchy, albeit as a mere ceremonial institution. They realized the power of the symbolism of the Imperial Institution. The retention of the monarchy in Japan I think had a calming effect on the post war reconstruction and democratization; that and having MacArthur as an American "Shogun" guiding them along. A modern example of how a monarchy has smoothed a transition from dictatorship to democracy is Spain under King Juan Carlos. In 1918, had the Allies permitted the Germans and the Austrians to retain their monarchies, the history of the 20th century may have been very, very different.
It was none other than Winston Churchill who argued "If the Allies at the peace table at Versailles had allowed a Hohenzollern, a Wittelsbach and a Habsburg to return to their thrones, there would have been no Hitler. A democratic basis of society might have been preserved by a crowned Weimar in contact with the victorious Allies.” in the aftermath of the Second World War (26th April 1946).
I think it's really hard to decide how much the downfall of the Hapsburgs and Hohenzollerns was due to internal developments before Versailles and how much was the result of the agenda of the victors. In the case of the Hapsburgs, I think a break-up of the Empire was inevitable but a Kingdom of Austria or even Hungary could have been salvaged from the ashes of defeat. Emperor Karl I was potentially a dependable ally for the victorious powers and had he been retained (and had he lived longer), he would have been quite a counterweight to the ambitions of those who wished to achieve union with Germany, I think, enabling Austria to transition to a full democracy.
The funerals of many ex-Monarchs in Germany in the 1920s, most notably those of Ludwig III of Bavaria and Empress Augusta, demonstrate that strong monarchist sentiment remained after the dissolution of the empire, supporting Churchill's argument.
Nonetheless, the Hohenzollerns pose more of a problem in that neither the Kaiser nor the Crown Prince could realistically be looked upon as reliable partners post-1918. Perhaps if the German Empire had been dissolved in favour of a return to a more rationalised form of the pre-1866 status quo, the result would have been a more stable, less threatening Germany with its Kingdoms and Grand Duchies providing a focus for loyalty and the reconstruction of a new identity. However, I doubt if this would have been acceptable to the German people themselves in the age of nationalism and the nation state and who is to say whether a fragmented Germany would have been more easily overrun by the 'communist' juggernaut, a fate that nearly befell Hungary.
I don't think Wilson "got" Europe - and he's not the last US President to have that failing. Advocating universal human rights is a noble cause but it doesn't necessarily follow that the ideal framework for their promotion can be the same in every culture or society. The US and the French pushed for the republican ideal as this corresponded with their image of democracy but, as we know, it's far from being the only one, and it could be argued that many of the surviving monarchies in Europe are more democratic than, say, the French or Italian Republics. Besides, as orrinhoover points out, although three defeated empires fell, the three victorious ones did very nicely out of the war, thank you very much, in terms of acquiring new colonies/mandates and/or greater economic/political influence.