Honestly, most of the monarchists of the period relied less on arguments for monarchy and more on arguments against the course of the Revolution. After all, it wasn't until the monarchy was fearing for their lives and tried to escape the country in the Flight to Varennes that anyone had made serious, well-received arguments for doing away with the monarchy in the first place.

Up to that point, so through the Estates General and the National Constituent Assembly, it was broadly understood that bringing France's finances into order would mean reform of noble privileges and a greater role of the people in the decisions of the state, but France was still an absolute monarchy. It wasn't until the Constitution of 1791, ratified after Louis attempted to flee the country, that France became a constitutional monarchy with a citizen king subject to the law, and it would still be almost another year before the king was deposed entirely.

So it's important to consider that France didn't enter this revolution attempting to do away with monarchy at all. It was successive waves of radicalization that even made the existence of monarchy a question. So until this point, the people we'd later identify as the hard-right were nobles who were more concerned about preserving their feudal rights and reinforcing the distinctions between them and the nouveau riche bourgeois members of the third estate's upper class. Old money vs. new money.

While there were people who believed in a republic within the legislative body, they didn't really have the influence or ear of the people until their influence started growing in 1792. Before then, the dominant current was with constitutional monarchists and liberal nobles who were reform-minded but not so radical as to overthrow their king.

It's not until almost a year later in the August 10 insurrection in Paris that the king is deposed and France gets on its way to becoming a republic. And he's put to the guillotine only a few months later. At this point most of the would-be royalists have fled the country. So when we're talking about royalists in the sense of people advocating for a French monarchy, we're mostly talking about people who are arguing for a restoration of the monarchy after it's been deposed, and they're doing so either from outside France or within certain territories in insurrection (chiefly the Vendee, an area in the rural west. There were other territories in open rebellion that wanted a republic but with more decentralized power so the provinces would have more influence than Paris.) Not only have the stakes raised, but the argument about the merits of keeping Louis on the throne have gone moot.

So the arguments you see in support of a restoration of the monarchy focus chiefly on the perceived failures or atrocities of the French Revolutionary government. A major sticking point is the Civil Constitution of the Clergy and the demands that priests adhere to it as civil servants of the French state rather than as part of the Catholic Church. It's such a sticking point that peasants who have no common interest with monarchists and nobles end up being the footsoldiers of what would come to be known as the Catholic and Royal Armies, the chief insurrectionists of the Vendee wars. These groups are further fueled by reports of the repression of territories in revolt and persecution of Catholics during the conflict. There was certainly a degree of freelance dechristianization (which was not according to official policy from the governments in Paris) but some of the accounts are almost certainly falsified or embellished to act as propaganda in the Vendee and Britain. This is where we start hearing about French Republican dechristianizers stripping nuns and priests, chaining them together and dumping them in the Loire, a supposed practice that came to be known as "Republican weddings."

From the side of the nobility, this is less about the administration of the church and more about reclaiming their lands (which, like lands belonging to all emigres were seized by the state and sold to new buyers.) They care less about the restoration of the Bourbons in and of itself and more about restoring the social and political order that came with the monarchy. And in the days of the right-wing backlash after Thermidor, it's apparent how little importance is placed on putting the next available Bourbon, Charles Phillipe, the comte de Artois (later Charles X) to the throne. There are royalists who plot against the Directory of post-Revolutionary France, but they're put down much more easily than the Vendee uprising ever was. The Directory was committed to recognizing the legal status of lands sold during the Revolutionary period and had no interest in restoring noble privileges, but nobles could return to being fabulously wealthy without the interference of the state, and that was far more a priority to many of them than actually restoring a king.

In total, you don't really see a lot of arguments for monarchy in terms of how good the system might be as opposed to others because the revolution moved so fast that monarchy was never really an issue until it was basically moot. The supposed monarchists after Louis is deposed represent a mishmash of interests whose only real commonality is opposition to the French Republic.

Hope that's helpful.