I guess such a supposition is a matter of
opinion rather then
fact each played it's own part in the battle against
Sauron but it is a matter of
opinion as to whether or not one played the "greater" part than the other, if indeed one side did play a greater part then the other.
Since this is a discussion of
the Third Age, I will deal with things that took place within the Third Age.
The
Disaster of the Gladden Fields was the starting point of the histories. With the disaster of the Gladden Fields, came the loss of
the Ring and the death of
Isildur.
The loss of the Ring is a important factor in the history of the T.A. Isildur was willing to hand the ring over to
Elrond and
Galadriel claiming his "pride had fallen". Would he really have given it up?
Gandalf is speculative, he claims it may have been better for Isildur to die rather then for him to live long enough for the Ring to take control of him. And even if he had given Elrond and Galadriel the Ring, what would they have done? Destroyed it? If so then they may have left
Middle-Earth due to the consequences of destroying
the one ring, and the resulting impotence of the Elven Rings, which they hadn't used yet. (At least not till the fall of Sauron at the end of the S.A.)
Or would they have become corrupted by the Ring? Would one of them take Sauron's stead as the new
dark lord/Queen/nymphomaniac?
If Isildur didn't march to
Arnor (He took this land as his own now, in place of his father
Elendil Voronda. Meneldil, his nephew was the
King of Gondor) then mayhap the fate of Middle-Earth would have changed for the worst. Already you can see that the existence of Arnor was important.
The
Witch-King also went to Arnor because it was the "weaker"
realm at that time. Well, if Arnor didn't exist then he would have no choice but to attack
Gondor. Gondor was eventually worn away by internal and external disputes whose to say the Witch-King couldn't wear away Gondor after a period of time?
Gondor of course acted as a buffer against the forces of
the East and South, but it sure wasn't a "stalwart soldier" in keeping out the
Easterlings and
Haradrim. The
Northmen were as important as Gondor in keeping away the threat from the East. Gondor of course didn't realise the importance of the Northmen buffer until it was too late, when they started to die out due to the great plague, in which nearly half of their population died. They also bore the full brunt of the assaults from the East, and after their defeat in
the Brown Lands (Together with the army of Gondor) they were scattered some joining their kinsmen if the North (
Dale) and some retreating West to remain in the Vale of
Anduin. A lot of them were reduced to servitude as the Waindriders took over and occupied their lands. Thus Gondor now had no bulwark to the N-E. but a enemy. King Caliemhtar, who had previously engaged in war to the south, which had been going on for some time, since the capture of
Umbar by hid grandfather Telumehatar Umbardacil. He then looked to the threat from the N-E. The Battle upon
Dagorlad was even, until Marhwini came to the aid of Gondor and the host of the Wainriders was scattered and they were utterly defeated, but some of the fled to the North. Now, Marhwini had earlier stirred up a rebellion and when he came back he found that although the Wainriders had retreated, most of the rebels were killed and he retreated back to the Vales of Anduin. The Northmen moved North after the destruction of
Angmar and the Fall of Arnor. They then saved Gondor with the ride of
Eorl. They were then given the Northern part of Gondor, 'Calenardhon' (named by them '
The Mark') and they acted as protection for Gondor for a length of time, until again saving them in the Pellenor
fields.
Of course the protection of Gondor may have encouraged the
Hobbits to move Westwards to the Vale of Anduin. The Hobbits moved up North though, when they were disturbed by the presence of
evil Men (The Wainriders) and the shadow in the forest. It seems by
Haldir's words that the
Silvan elves knew something of Hobbits, or at least had encountered them, but oddly
Treebeard has never heard of Hobbits and was mystified by their appearance, though he does note that their voices reminded him of something, so Treebeard may have missed the mass immigration of Hobbits, though it may not have gone unnoticed by the Northmen and the
Elves of
Lorien. Yet, the Hobbits played one of the most important parts, if not
the most important part in the war against Sauron, the finding, re-finding and destruction of the Ring-achieved by Hobbits, though of course there were other factors. But how did the history of the Hobbits work out? It is certain that Arnor played the more explicit part in the history and movements of the Hobbits.
They provided as safe-haven for the Hobbits in the early settlement in
Eriador and later gave them the '
Shire'-a important factor in the war against Sauron. More on that later.
The war in Arnor, though troubled some of the Hobbits. The
Stoors moved south, to
Dunland, but excavated back North and eventually crossed the
Misty Mountains and resided close to the Gladden Fields. This is a important part in the development of the affairs in the T.A. The removal of the Stoors back over the Misty Mountains lead to the finding of the ring by Deagol and then
Smeagol's taking of the ring. Assuming they had not found it it may have eventually been found by Sauron.
Arnor's importance is tied up with Hobbits and thus the ring, for all their victories in
wars, Gondor's efforts were in the end futile but for the destruction of the ring, in which Arnor played a important part.
Arnor also kept close ties with the Elves. Gondor lost most of it's ties with the Elves, but Arnor still communicated and allied itself with
Rivendell and
Mithlond. Arahael and his heirs were all fostered in Rivendell by Elrond and this the final strain of descendants of Elendil were also fostered by one of the wisest of all the Elves in Middle-Earth and had a teaching and upbringing and nobility that far surpassed any
Gondorians. In. In Arnor the line of
Kings endured, culminating in the saving of Gondor on the Pellenor fields and the re-uniting of the line of the
Half-Elven. If it wasn't for
Aragorn then Sauron would have got the ring.