It's not very likely Logain "saves" Rand.

IMO there will be two phases to the Strike. In the first one Nynaeve will use Callandor (in a link with Rand and Moiraine) to remove the residues of the sealing that would prevent the Bore from closing.

In this phase, Rand's soul will be translated by SH into Moridin's body using the bond between Rand and Moridin. Alivia will be hidden by the ter'angreal that shields from Shai'tan's attention and she will kill Moridin, freeing Rand.

The second phase will involve Rand in the afterworld with the Heroes of the Horn, holding Justice against Shai'tan to distract his attention. Meanwhile, Nynaeve will remove Shai'tan's "tentacles" holding the Bore (like the taint holds a brain). In tel'aran'rhiod the Dreamwalkers will help the Wheel speed up the process of reverting the Pattern to its normal state: a "weakness" which is how decay and death can exist in Creation, not a Bore letting Shai'tan actively touch the world. I'm convinced for many reasons Lanfear and Beidomon created the Bore in TAR, because I think what became "the Bore" used to be the threshold between the living world and the Afterworld. It's through that weakness the Wheel can use a trickle of Shai'tan's power to produce the decay to death process.

It doesn't matter much that Rand has to be seen as "evil" for future legends. We have a poetry from the 4th that basically worshipped him.... The next age isn't close from "our age", which happened before the AOL or close to that. Brandon has already confirmed the next Age isn't the one in which channelling disappears.