Sealing the Bore will require Rand, Mat, and Perrin because each is ta'veren. It is highly unusual for even one person to be ta'veren, and even more unusual for there to be more than one. The WoT series has revolved around all three characters, not just Rand, and both Mat and Perrin have developed unique and compelling skills in their own rights that merit consideration in the sealing of the Bore.

Short version of theory: Perrin finds the Song, Rand teaches it to the Aiel, the Aiel sacrifice themselves to 're-grow' the Pattern over the Bore by singing the Song, and Mat (and Rand) provide cover from the Shadowspawn.

Long version:

1. The 'Last Battle' is not the last for humanity or the Pattern, but the last for the War of Power

The War of Power never formally ended and won't until the DO's prison is fully restored to pre-Bore strength. This being only the Third Age, it does not make sense for the Last Battle to be the end of humanity or of the Pattern -- there are four more ages before the wheel completes a full turn, and even after the Seventh Age the cycle merely repeats itself.

At the end of the Second Age, the seals put in place by LTT and the Hundred Companions simply limited the DO's reach and temporarily imprisoned the Forsaken. The war itself did not end with that moment. Perhaps it might have eventually, had saidin not been tainted and the male Aes Sedai not gone mad. Instead, the Breaking of the World served to disrupt and suspend active military conflict between the forces of the Light and the Dark.

In the three thousand years since then, both sides were either waiting for or actively preparing for the Last Battle. Darkfriends proliferated, the Blight grew, and active conflict occasionally broke out, such as during the Trolloc Wars. The seals on the DO's prison gradually weakened. The Forsaken eventually escaped.

And mostly importantly, the Dragon was reborn. Not because every age has a dragon, per se, but because LTT, despite his enormous talent and power, had failed to restore the Pattern to its pre-Bore state. (After all, one could argue that the Bore is the most important moment across the entire Wheel of Time and its 7 Ages, given that it signifies a rupture in the Pattern itself.) As LTT said to Rand atop Dragonmount in TGS, the Dragon was born again for another opportunity to 'get it right' -- i.e., seal the Bore, win the War of Power, and restore the Pattern to the way it was made by the Creator.

2. The Real Seal is not from the TS or the TP

For all their wisdom and ability, the AoL Aes Sedai never figured out the nature of the rip in the DO's prison and how best to restore it. Decades passed before they even knew what the Bore had done. And the War of Power raged on for years before LTT and the forces of the Light realized they needed to seal up the DO in order to achieve military victory over the forces of the Dark. This perhaps is a function of the complexity of the problem and the difficulty of doing any real research -- after all, there was a war on, and it's not like people could go take readings at Shayol Ghul.

So it is of little surprise that Aes Sedai only attempted to seal the Bore late in the war, and when the Light was near the brink of defeat. And it is even less surprising that they struggled to reach consensus over how to do it -- with no one anticipating the DO's counterstroke.

RJ and others have noted that it is not clear what might have happened had both saidar and saidin been used to create the seals -- would both halves of the TS been tainted, or would it have fully restored the prison? Given the artificial nature of the seals, it seems like even both halves of the TS would not have been a permanent solution and in any event it seems like it would have resulted in both getting tainted. The plan to build a 'new' prison with the KCs was also temporary in nature -- as it was meant to simply hold the DO back until a permanent fix could be found.

So if not a product of the TS, what can close the Bore? Many have argued the TP must be used, citing the "three shall be one" prophecy and Rand's ability to channel the TP. There are a lot of compelling reasons why this makes sense, including the odd flaw in Callandor and whether this might enable magnified TP channelling. But this means that the DO's own power, which the DO allegedly controls access to, is to be used against him. This also means that the prison, which was made of the Pattern and by the Creator himself, is somehow capable of being plugged by the very thing it was constructed to contain. Granted, there is a bit of RJ-like irony here that may prove irresistible, but it is not logical.

3. If not the TS, and if not the TP, then what?

Here is an outline of what I think each of the three ta'veren will do in the MoL endgame that leads to the sealing of the Bore and the final end to the War of Power:

A. Perrin

Perhaps aided by the wolves, Perrin discovers the Song. It may be locked in a ter'angreal someone discovers or maybe he accesses it in TAR. He teaches it to the Tinkers, and maybe the Ogier, and they bring this to the Last Battle.

B. Rand (and the People of the Dragon)

Rand relays the Song to the Aiel, reuniting them with the Tinkers and explaining to them that the Song is needed to defeat the Blight and 're-grow' the Pattern over the Bore, thereby restoring the DO's prison to full strength.

After all, if the prison was originally built by the Creator outside of the Pattern, and if the Bore is a rupture in the Pattern itself, then fully sealing the DO means the Pattern has to be 'regrown' over the hole the Bore made. The Song seems to tap into a different kind of power to effect the growing of living things.

What is needed is for Rand to 'clear away the rubble' of the LTT seals, and perhaps remove the Bore itself, if the vessel Lanfear made to open it is somehow still there, holding it open. Perhaps this is where Callandor comes in, as well as Rand's use of the TP to break the remaining seals. Then he steps aside, and the Aiel come forward.

Here is where a number of things we saw very early on in the WoT series come into play: the Way of the Leaf, the Aiel as the 'People of the Dragon', the Da'Shain Aiel as servants of the Aes Sedai (and especially of LTT), the Aiel's Covenant and the breaking thereof, and the Aiel as a disciplined and courageous people, not just in the Third Age, but even as the peaceful Da'Shain Aiel (see, e.g., how ten thousand Aiel linking arms and singing before a mad Aes Sedai Jaric Mondoran were killed nonetheless, down to the last person. TSR).

But I think, since these Third Age Aiel have wrought great violence in their lives, it is necessary for them to die in Singing the Pattern closed over the Bore, in order to ensure its purity, not unlike the clean pool of saidin was created. Indeed, it may be that there deaths are necessary, given the precept of the Way of the Leaf that violence harms the one who does it as much as who it is done against (TEotW).

This is why, in the Fourth Age, the Aiel are all but gone, with only a remnant of a remnant surviving.

C. Mat

All along, and after the DO's prison is sealed, there are a lot of Shadowspawn and Forsaken to clear away. This is where the greatest living general comes in, taking command of the combined forces of the Light, including the Seanchan, to prevent the Aiel and Rand from being killed before the prison is Sung shut.

4. Concluding Loose Ends

There are obviously a lot of loose ends from the prophecies that need to be addressed, such as twice dawns the day, and Avilia helping Rand die. How Moridin, Fain, and Shadar Haran are dealt with also remains. But nothing postulated here is necessarily inconsistent with these events coming to pass in some form.