Catholics are free to be either republican or monarchist. As for the British state, it is a republic implicitly since the Glorious Revolution.
The French Revolution was basically an attempt to import the British constitution to the continent. Voltaire and Montesquieu were ardent Anglophiles, became Freemasons in London and wrote books extolling the virtues of the British constitution (the Glorious Revolution and Magna Carta).
Republics are simply “public things” and, as such, can have a monarch as head of state.
The French Revolution may have started as you say, but it quickly degenerated into a type of communist and murderous revolt replete with dictators who saw themselves as demigods.
The French Republic was as anti-Catholic and militantly atheistic as any totalitarian state of the 20th century. French republican guardsmen would travel the countryside and line up villagers to come forward and spit on crucifixes held upside down and the like. They turned churches into “Temples of Reason” replete with “festivals in honour of Reason and the Virtues” - including Notre Dame Cathedral in Paris.
If you doubt any of this, just pay a visit to the St Genervieve’s church in Paris, the national whatever they call it. There where the altar once stood is a large monument to France’s soldiers. In the middle there is a ball tied to a thin wire attached to the cupola above. The ball continually moves - a way the French have of constantly telling the Church that science trumps religious belief. And in the basement there are all manner of tombs to the “republican atheist saints” of the Revolution, together with flowers and even candles. There is even a video shown over and over of the burial of Victor Hugo where it shows how closely it resembles a church procession to a saint. Don’t take my word for it - go and see this for yourselves the next time you are in Paris.
It was France that paid lip service to the ideals of its Revolution and later returned variously to imperial, royal and republican forms of government. (This is why France has no pledge of allegiance - they never know what kind of government they will have next.)
In 1893, the French parliament decided to offer the crown of France, once again, to the head of the Bourbon Royal House - but he said he would only accept it if the country got rid of its revolutionary tricolour and returned to the old Royal flag of France (yes, a stupid move on his part).
The “spirit” of monarchy has never left France (which recognizes still the aristocratic hierarchy of bygone years). Its president carries himself like a king (and is referred to in the French press as the “President-Monarche”).
Republicanism has produced only a few true democracies and the U.S. has been in the habit of supporting republican heads of state who are little more than despotic dictators (Mubarak is but one example). And the U.S. took down the Shah of Iran (a great friend of the U.S.) only to have him replaced by . . . we know the rest of the story and the legacy of President Carter. Such republics put the “mock” into “democracy.”
Monarchies tend to be conservative, spiritual and economically stable places. The U.S. has made many, many errors of judgement in terms of foreign policy on this score. Hopefully, the U.S. will learn a few things.
The Catholic Church is also a monarchy and I don’t think anyone envisions a republic for the Church either.
Alex