I feel European monarchies were ready to fight against an Empire, but not to fight an imperial Republic.
For example, if the French went into a republican crusade through Europe (analogue to the Napoleonic Wars), the resulting client republics would not only enjoy the full support of a sector of their population, but would also have an incredibly high morale when in war against the reactionary enemies since the survival of their (republican) state would only be accomplished if they were ultimately victorious. Thus there would be no changing sides when things go bad for republican France, since only by supporting them could they assure their own survival.
The Republic would provide France the ability to really destroy the balance of power and the status quo, and create a new revolutionary republican order in Europe.
OTL Napoleon abstained himself from dismembering Prussia or Austria (which he had the chance to do at various points), since that would shake the balance of power too much and could have provoked his allies-in-turn to turn against him.
But then again this cannot happen if he completely erases away each enemy after defeating them, and replaces them with Republican Confederations that would depend on his victory to survive: Thus an hypothethical Prussian Republic/German Republic would fully support the dismemberment of Austria in the way to create a new order, and the spanish liberals could sustain a Spanish Republic without the unrest caused by a foreign monarch imposed of them.
Before crowning himself, Napoleon had been playing outside the system, and the system was glitching because of the anomaly it couldn't process or understand. However, when Napoleon became Emperor, he entered the system and the system was able to understand the situation better. The thing is he continued to fight the system after that when he was a part of it, and with unpopular things like putting his relatives on the thrones, etc. the monarchies of Europe were able to do what they had been doing for centuries (political-dynastic intrigues, etc.) and precipitated his fall.
As another example, was the British willingness to recognize the Kingdom of France defunct and the French Republic as the legal government, but then officially declaring the French Empire illegitimate and Napoleon a usurper.