And the soccer national team demonstate it, right?
Yes and they were all "guillotined" (figuratively, not literally) for it
To expand on my prior points: the monarchist groups in France are a minority, but they are a minority whose ability to influence things socially and politically is not always seen but very much acknowledged by the French establishment.
The change from First Republic to First Empire made no difference to the fact that Napoleon was the man in charge, even though both systems were constitutional on paper. The Bourbon Restoration of 1815-30 did not restore absolutism, lessons had been learned from the Revolution. The Orleanist "July Monarchy" was more liberal in character.
The Second Republic became the Second Empire under Napoleon III, and once more you saw a familiar pattern: constitutionalism on paper, a somewhat militaristic state in practice. Ironically, the forces that defeated the Empire and led to the final demise of monarchy in France, would be the same forces defeated in WWI.
During the Third Republic, there was a widespread assumption that it was only a temporary solution before some form of monarchy would be restored. The chance of it was quite high during the presidency of MacMahon, but after he left office the movement declined even though the monarchy v republic debate (and the internal Legitimist v Orleanist v Bonapartist one) remained part of political discourse until World War I at the latest.
World War I was a traumatic experience for France, which profoundly affected its society, politics and position in the world. In practice, the "monarchy debate" took back stage to the fact that not only the Third Republic but democracy in general was under threat from new, extremist movements. And then there was World War II. There's no doubt that the failings of the later Third Republic (basically the interwar years) and the Fourth Republic, where chronic instability ensued, may have been seen by monarchists as an opportunity for restoration that was missed.
It wasn't to be but what De Gaulle created was a Fifth Republic whose President was the most powerful head of state since the last monarchy.
Even though that won't happen, there's no doubt the monarchist movements in France remain active and continue to wield influence however subtle in modern French politics. The "monarchist vote" is a bloc that politicians do silently acknowledge as having potential to influence things. They can integrate themselves into whatever base they find suitable.