Napoleon's France and the Coalition are two forces that's almost impossible to coexist.
France post-1789 was a state born from very radical ideas; for the coalition, it was a poison they need to neutralize, otherwise it'll also kill them later on, with or without war.
As for Britain, it presented a shocking surprise. France after Louis XIV was on a decline, and the British knew then that they will never be a serious threat to London again. Until the French started kicking the other great powers' collective asses big-time in the wars that came after the Revolution.
This fact single-handedly ruined Britain's long-time foreign policy, and thus they acted immediately before it gets out of hand.
I disagree with you on several points.
Of course, you are right on the fact that French revolutionary ideology was a scarecrow for the aristocratic ruling elites of Europe.
But, this was past to a large extent at the time of Napoleon. Napoleon was a stabilizer. He proclaimed himself the end of the French revolution was over. Napoleon was a soldier, a man of order who despised mob violence and who favoured property and merit. All his strategy was stabilizing his country that had been upset by the previous regimes (both monarchic and revolutionary regimes) that had failed to run the country and fix up what had been blocked and had derailed in the government and in the laws.
And most of all, it was an accessory reason although a real concern, but it was not the main and truest reason.
It was placated for political conveniency : the so famous “we don’t fight your country but your evil leader and his evil ideology”. 2 centuries later, there is still a great many people in France who admire Napoleon because he elevated France to its highest level of power and glory, before ruining it all.
The deepest truth it that regimes and ideologies matter little. If you need proof for it, just consider the fact that, except the expropriated habsburg lords who needed time to accept their loss, nobody cared about the kind of anti-noble “democratic” ideology of the swiss cantons. And, although Louis XIV’s France was the archetype of a monarchy that favoured the nobility and other privileged groups, most of Europe entered in coalition against monarchic France.
What really mattered was and has always been geopolitics, balance of powers and competing imperialist ambitions.
And even before the French began considering exporting their revolutionary ideologies, emperor Josef II of the HRE tried to profit from the weakening of the United Provinces and of France while these countries faced internal turmoil.
France after Louis XIV was not in decline. It is often ignored that French economy and French trade grew faster than the british economy in the 18th century until 1789. It was one of the reasons why Britain’s merchant elite wanted to go for the French jugular in the war of Austrian succession and in the seven years war. They were afraid of French competition. Britain really took off from the 1770’s on, when it became able to boost its economy and its finance with massive injection of indian steroids. And if some people in Britain had ever forgotten that France still was a real strategic threat for Britain, they had the American revolution war to remind them that France was indeed the main threat to Britain. The british ruling elite was in fact perfectly aware of the French threat as a strategic rival. They knew that the French Navy was at ont of its all-time peaks under the reign of Louis XVI.
The French revolution caused a real crash in the French economy. One of the reasons often unknown for Napoleon’s popularity among his contemporans is that he restored order and created the conditions for the French economy to boom back after the revolutionaries had caused it to go bust.
Britain’s foreign policy was not ruined by France’s resurgence under Napoleon. Britain always stuck to its century long strategy that was balance of powers in continental Europe and avoiding France becoming too powerful on the continent and too powerful a competitor overseas. Britain did not like competition nor competitors. It prefered monopoly which was far more lucrative.