The British, various German states to an extent adopted a rifle for issue to some light infantry units. The most 'famous' rifle of the period was undoubtedly the British Baker Rifle, but it was by no means the first rifle produced and employed during the period.
It was preceded by the excellent short German rifle which armed the German jager units in the War of the Revolution in North America (1775-1783). The Americans produced the Kentucky/Pennsylvania long rifle and rifle units were raised, organized and employed in the war with Great Britain and faced the jagers on the battlefield. The Americans respected and feared the German jagers, who were excellent troops and learned that if they employed riflemen, they had to be supported by musket and bayonet armed infantry. The rifles of that war could not be used with bayonets and they were slow to load. That made those units vulnerable to sudden bayonet attacks.
The predecessor to the rifled long arm in the British service is probably the famous Ferguson Rifle, invented by the officer of that name who was killed in action at King's Mountain in 1780. It was an excellent weapon, was a breech loader, expensive to produce and was not adopted.
The question being put is, however, why didn't the French army of the Wars of the Revolution and later the Grande Armee use the rifle and employ rifle-armed light infantry?
French light infantry evolved into an excellent arm and their ability to fight in open order on both offense and defense was part of the French tactical system that evolved from the defeats of the Seven Years War and was the subject of long debates, experimental maneuvers, and finally battlefield necessity.
It should be noted that no army of the period used riflemen better than the British Army and that no army had an overwhelming number of rifle units. The rifles of 1792-1815 were still slow to load, but the British had solved the bayonet problem by designing the Baker Rifle to take a bayonet and designed a specific bayonet for that purpose.
The French did design and manufacture rifle long arms and pistols, but the average French infantryman preferred his musket. Generally rifled weapons, especially long arms were not popular with the French infantryman. The rifled carbine (carabine d'infanterie) that was issued to officers, sergeants and fourriers of voltigeur companies was an excellent weapon, the Model of the Year Twelve. It was relatively short (40 inches overall), weighed 8 pounds, fired a twenty-eight to the pound forced ball, was slow to load, was difficult to load and had no bayonet.
There was also a 'very short' rifled carbine manufactured at Versailles from 1793-1800 and was sometimes carried by officers. Rifled pistols were also manufactured and carried, Coignet is noted as buying and carrying them. There was an armorer in Ratisbon named Kuchenrenter who manufactured rifled pistols, cost 150 francs a pair and used a forced ball.
In the end, for the French, rifled firearms were not adopted for the army or a portion of the army probably because they were just not popular.
References that might be helpful are:
British Military Flintlock Rifles 1740-1780 by De Witt Bailey.
The Book of the Continental Soldier by Harold Peterson.