It would matter on who would call them "Elite" units. The British army certainly did from the early 18th century and then on, especially during the Seven Years War. Their martial culture set them apart from other British soldiers of the era. Many English officers recognized their abilities and knew it would come to use, so did other foreign armies such as the Dutch and French, and created regiments for them to gather in. Not only were they allowed their tartans and claymores in these regiments but most of their officers were Gaelic speakers creating a clan like setting among the regiment. By Culloden Highland regiments had a reputation of being fierce fighters and and excellent troops to use for the British Army.

They also tended to be part of the fiercest fights and battles. The 43nd Regiment of Foot, later the 42nd, (Didn't become the Black Watch under 1881) fought in Fontenoy and showed their prowess as disciplined troops. And then Ticonderoga and Fort Duquesne in the Seven Years war, suffering many casualties. Post-Culloden showed their excellence in warfare due to the banning of tartan and weapons in Scotland. By allowing Highlanders to wear tartan, fight with weapons, but for the King, they gained loyal subjects. There was also a patriarchal bond with their officers, more so if the officer spoke gaelic, shared familial relations, and treated their soldiers well. The Crown now had a major manpower resource to use with many Highlanders willing to join the army to wear their tartan again, but also created a source of employment for them.

Most people see Highlanders as berserkers with claymores, but that maybe a slightly distorted image. They were indeed known to have high spirits during a battle but were professional and able to control their emotions. At times their spirit overrode their discipline but that was not always the case. They had the training that the other soldiers had such as rolling fire and volley fire, but they did retain their Highland charge with broadswords and dirks. The Highland charge was effective against regular troops but not so much against Indians and bushrunners who put distance away from their blades during the Seven Years war as noted by some officers. The charge remained to be part of the Highlanders' psychological warfare that they carried through out the wars in the 18th century.

Highland regiments gained a lot of praise from their fighting in the Seven Years war, and were eventually accepted as part of their army by many English officers. Those were the formative years for Highland regiments to have gained their reputation as "Elite" units.

Follow up question: this seems to exist in most militaries around the world, one group being the best of the best. Is there a point in history where we begin to see high specialized military units begin to develop, aside from progression of the citizen army to a professional one?