Edit: I feel it is better for the Nords to stop the Dominion in southern Cyrodiil than stop them in southern Skyrim. Better for both Cyrodiil and Skyrim.
Yeah, but thats the Akaviri. They were a far more capable fighting force than an army of elves that have a reputation as being among the worst melee fighters on Tamriel.
And indeed, it would be. But depending on how the Civil War is decided (and what happens immediately afterwords) Skyrim will either be have to fight in its southern mountains (or perhaps northern Cyrodiil) or will have to rush to have forces ready in Cyrodiil. Especially if the Dominion decides it wants to attack the moment the war ends. Either way though, Cyrodiil is likely going to be devastated again, and I don't think theres going to be any way around that unless the Legion both wins the civil war and miraculously returns to its former strengths.
We also have the issue of Hammerfel. If they were to be called upon for reinforcements (and lets face it, they will eventually have to come into the war with the Dominion again anyway. Especially if the Dominion somehow manages to take High Rock and Skyrim) Skyrim would be far more likely to get them to agree to it than the Empire would be.