France declared was on most of Europe during the "constitutional monarchy" phase of the Revolution. Louis XVI signed off on it. The flight doesn't change this. IOTL, the remaining European monarchies were remarkably lackadaisical in fighting the 1790s wars even with the execution of the King and Queen of France.
The French Republic is more united internally by not going through with the execution. Though the flight to Varennes didn't make the execution necessary, they could have kept the royal family imprisoned somewhere or just sent them into exile. There was no danger of counter-revolutionaries freeing them, unlike with the Russian royal family, and anyway as bad as the Jacobins could be, they weren't Bolsheviks. The execution of Charles I of England really hurt support for the Commonwealth, though the effects didn't take hold until years later. Charles I acquitted himself much better at his trial and execution than did Louis XVI, but the latter retained enough dignity that his execution was really something the Republic could have done without.
Incidentally, this lesson was learned. Nikolai II and his family was the only instance of the execution of a deposed European monarch after 1793. If you really want to stretch things, you can count Maximilian of Mexico and the assassination of the Portuguese royal family. But everyone was pretty much sent into exile. Even the Bonapartes and Wilhelm II. Even Stalin didn't execute the kings of Bulgaria and Romania.
The parallel is with James II and his descendants, who went into exile, caused some trouble, and eventually faded away.
What the continued existence of Louis XVI does is to present a political problem for the Coalition, assuming they defeat Napoleon on schedule. The problem was that he simply wasn't a very good king. Most likely they will persuade him to step aside in favor of his son, but if for some reason that can't be done, re-establishing the Republic or continuing the Empire under a Napoleon II under tutelage become more attractive options than IOTL.