Even if Louis XVI hadn't left, I'm not sure he would have been spared by the Revolution.
The Revolution wasn't doomed to evolve in a really precise way. Louis fleeing to Varennes represented an important change in mentalities, for what matters to the king and as well to the assembly (critically with the ludicrous official tale about the king being kidnapped).
Without Varennes, republicanism would have an harder time imposing itself and Louis remaining a token figurehead; inner restrictions on movements would probably be delayed; and Feuillants holding more power and critically, no
Champ de Mars massacre that radicalized Jacobins and decredibilized Feuillants.
It doesn't mean revolutionary violence would disappear but it would be turned to other focuses.
He was more of a burden to the Revolutionnaries than an asset: remember that he was called "M. Veto" at the time because he kept using his veto on the laws established by the Assemblée Nationale.
First, it's worthwhile noticing that there was not National Assembly in June 1791, and that the National Assembly, and the constitution, appeared in September. After Varennes. I'm not that sure arguing of something happening in September is wise when it comes to an event in June.
That Louis demonstrated his opposition is a thing, but the royal "strike" mostly happened after June.
While convenient, treating the National Assembly as an unified body is probably not really useful historically. The diverse parties in presence had different views and objectives, Feuillants, Jacobins, Démocrates (in the old sense of "populists", "agitators", "leftists") without counting Marais that gathered the majority of the assembly and without real defining ideology (and usually served interest of the most popular party).
Again, without massacre of Champ de Mars, Jacobins would radicalize more slowly, and probably have inner divisions between their center-right and left appearing more obviously.
There is also the whole problem of the Armoire de fer and what it contained. If word ever gets out about it, Louis could be in serious trouble. The armoire did contain a number of compromising documents after all, that were used in Louis' trial.
It's to be noted that it wasn't discovered before late 1792, mostly because it wasn't even a thing before being build in mid-1792, and by denounciation in a climate where royal figure was not only discredited but as well being overtrew by the proclamation of the Republic.
Sure, you probably had a few villages burned and a few people slaughtered in the past during rebellions, but not on that scale. And it certainly wasn't considered to be the normal procedure... Among the authorities, there really was the idea of conducting a purge in Vendée.
The official estimations, in lack of proper statistics, were heavily based on wishful thinking from both monarchists and republicans, each side tending to blow out numbers in order to point monstruosity or determination of republican side.
Around 150 000 deaths (including non-Vendeans and including undirect causes, as starvation, diseases
WITH the republican Vendeans death) for the whole Vendée militaire (meaning more than just Vendée strictly speaking), with a similar death rate among Republican troops points an harsh and bloody war, but not really an obviously unbalanced conflict (if the length of the war didn't pointed out that already).
That the idea existed among some circles (I'd tend to think that the Assembly was divided as it always was on it, Enragés tending to blow everything out, Robespierristes and Indulgents being themselves divided between giving room to popular opinion and radicals to vent off their rage; participating actively or not caring much).
Without an unified political rule, the revenge fantasy of radical terrorists (in the political sense) had an hard time making it a thing (and if we have to use the comparison with Fouché's treatment of Lyon, or Trueau's destution in 1794, may have importantly weakened their position if really tried).
Again, treating the Assembly and political power as unified in political rule is a really heavy mistake, because this desunion is another explanation : the army itself was poorly managed and the conflict was often the war between plundering Vendeans (they weren't the last ones to plunder and raid non-Vendeans (the definition blurry at best) places and plundering Republican troops.
There are even a few historians of the Wars of Vendée that are ready to use the terms "Genocide" or "Populicide" to talk about it. It's a big controversy to use such terms, but if people are willing to use them for what happened in Vendée, then I don't think you can say it was standard for the time.
Allow me to intervente, there. The qualification of "génocide" is controversial because the people using it are usually politically motivated : Reynald Secher is a catholic hardliner and close to Action Française (far-right monarchist political movement), for exemple.
Pierre Chaunu, former chronicler in Radio Courtoisie (far-right radio station).
It doesn't automatically led to a refusal of their thesis, but pointing out that the main proponents of these are either far-righters or close to them is still relevant to the whole discussion, critically giving that the whole crushing masse of historians disagree with them on this.
It's essentially a french issue, meaning there that past a vocal minority in France, you have none to very little support for these thesis internationally among historians.
And of course, it's not because one is running circles shouting "Genocide" that it gave the concept used as such any credibility.
Hell, even François Furet, someone that it would be really hard to paint as a leftist, argued in favor of distinguishing crimes, slaughters from a genocide.