In the LOTR Fandom wiki article for the Grey Company, it says:
Led by Halbarad Dúnadan and the twin sons of Elrond, Elladan and Elrohir, these Rangers were considered the best mortal fighters in Middle-earth.
Anyone can edit this site so it shouldn't be taken too seriously. However, I thought it was an interesting claim. Is there any evidence for/against this claim?
While I struggle to find a direct quote suggesting that the Grey Company were considered the best mortal fighters in Middle-earth, there are a few instances in which their prowess is mentioned as well as others which describe the Rangers of the North in an almost legendary fashion.
Firstly, specific to the Grey Company, we have the following from Theoden after the arrival of the Company in Rohan:
Théoden was indeed glad of the news. 'It is well!' he said. 'If these kinsmen be in any way like to yourself, my lord Aragorn, thirty such knights will be a strength that cannot be counted by heads.'
The Return of the King, Book V, Chapter 2, The Passing of the Grey Company
While this is the only relevant quote I can find specific to the Grey Company, throughout the stories the Rangers of the North are often lauded by wiser characters (i.e. Gandalf, Tom Bombadil):
Then he told them that these blades were forged many long years ago by Men of Westernesse: they were foes of the Dark Lord, but they were overcome by the evil king of Carn Dûm in the Land of Angmar.
'Few now remember them,' Tom murmured, 'yet still some go wandering, sons of forgotten kings walking in loneliness, guarding from evil things folk that are heedless.'
The hobbits did not understand his words, but as he spoke they had a vision as it were of a great expanse of years behind them, like a vast shadowy plain over which there strode shapes of Men, tall and grim with bright swords, and last came one with a star on his brow. Then the vision faded, and they were back in the sunlit world.
The Fellowship of the Ring, Book I, Chapter 8, Fog on the Barrow-downs
'Do you really mean that Strider is one of the people of the old Kings?' said Frodo in wonder. 'I thought they had all vanished long ago. I thought he was only a Ranger.' 'Only a Ranger!' cried Gandalf. 'My dear Frodo, that is just what the Rangers are: the last remnant in the North of the great people, the Men of the West. They have helped me before; and I shall need their help in the days to come....'
The Fellowship of the Ring, Book I, Chapter 11, A Knife in the Dark
Both Tom and Gandalf seem to hold high esteem for the Rangers (rightfully). Tom in particular induces visions of great heroes in the Hobbits, Tom's description is the first of the humility and isolation of the Rangers from the rest of the peoples.
Aragorn himself compares the Rangers to the Men of Gondor, informing Boromir of the acts they carry out without praise or glory:
Many evil things there are that your strong walls and bright swords do not stay. You know little of the lands beyond your bounds. Peace and freedom, do you say? The North would have known them little but for us. Fear would have destroyed them. But when dark things come from the houseless hills, or creep from sunless woods, they fly from us. What roads would any dare to tread, what safety would there be in quiet lands, or in the homes of simple men at night, if the Dúnedain were asleep, or were all gone into the grave?
'And yet less thanks have we than you. Travellers scowl at us, and countrymen give us scornful names. "Strider" I am to one fat man who lives within a day's march of foes that would freeze his heart or lay his little town in ruin, if he were not guarded ceaselessly. Yet we would not have it otherwise. If simple folk are free from care and fear, simple they will be, and we must be secret to keep them so. That has been the task of my kindred, while the years have lengthened and the grass has grown.
The Fellowship of the Ring, LoTR Book 2, Ch 2, The Council of Elrond
In this snippet from Aragorn, the Rangers are described as humble servants of the People of Middle-earth. Willing to do things so that the majority of the people may live in blissful ignorance and happy to do it in secret if it is required. This description alone sets them greatly apart from the Knights of Gondor, in particular Boromir, who seem to be more interested in glory. Although, it must be noted that, the Rangers of Ithilien reflect the Rangers of the North in some ways, surviving the wilds to keep evil at bay.
Returning to the Rangers of the North, Breelanders described them as fantastical beings with improved sight and hearing:
[In] the wild lands beyond Bree there were mysterious wanderers. The Bree-folk called them Rangers, and knew nothing of their origin. They were taller and darker than the Men of Bree and were believed to have strange powers of sight and hearing, and to understand the languages of beasts and birds. They roamed at will southwards, and eastwards even as far as the Misty Mountains; but they were now few and rarely seen. When they appeared they brought news from afar, and told strange forgotten tales which were eagerly listened to; but the Bree-folk did not make friends of them.
The Fellowship of the Ring, Book I, Chapter 9, At the Sign of the Prancing Pony
Finally, the following example seems to suggest that if the Rangers were unable to find a permanent path, it was truly a challenging place to be wandering. Again describing the Rangers as the upper limit of skill and prowess:
The marshes were bewildering and treacherous, and there was no permanent trail even for Rangers to find through their shifting quagmires.
The Fellowship of the Ring, Book I, Chapter 11, A Knife in the Dark
It depends who you ask and what exactly defines being the "best fighter".
The Ranger's effective service - and indeed their survival in the years of Sauron's advance - depends on their true nature and capabilities remaining hidden. Most people either do not know about them or (like the Bree-landers) view them as untrustworthy vagrants, not heroic defenders.
There are people who probably do consider the Rangers to be the best mortal fighters:
The Rangers themselves
Elrond who knows them well but is quite prejudiced against other Men:
But in the wearing of the swift years of Middle-earth the line of Meneldil son of Anárion failed, and the Tree withered, and the blood of the Númenoreans became mingled with that of lesser men.
This establishes that he helds both non-Númemoreans and Gondorians lesser than northern Dúnedain
But others in universe do not: Probably the Easterlings and the Haradrim all have their preferred champions who they deem superior, the Swan Knights of Dol-Amroth also stand a chance, and many (Bergil for example) would put the Citadel Guard in the first place.
And that is without considering other races that are mortal but are not Atani: Dwarves and orcs.
How good can the Rangers really be?
I would think that their skill is indeed exceptional in their chosen role of "stealthy light infantry (but also doubling as light cavalry)"
They can effectively control large territories in Eriador a much larger force with a lot of high-profile fortifications could not. They track and intercept raids of orcs, trolls and other enemies.
But this is not only way of War, nor the only one needed: No doubt the people of Pelargir and Umbar would beat them on the sea, the Swan Knights are probably better as shock heavy cavalry, the Citadel Guards as stand-your-ground heavy infantry, dwarven warriors in clearing out underground orc-holds.
And though they are more barve-and-well-rounded than extreme professionals the House of Eorl also has some pretty impressive feats: They have slain Scatha and the Witch King - who had already destroyed Arnor as a kingdom and recently routed Rangers at Sarn Ford.
