Well, the Civil War only exists because of previous dragonborn. Might as well make the new one involved. First, it's a war over Talos obviously. Secondly, Martin. Martin left a void in the empire after his death - he warded off one crisis, only to accidently create a new one. In the form of Titus Mede. Titus Mede was a random warlord who saw how weakened the imperial city was after Oblivion and took it over with a measly 1000 troops. And started playing "pretend dragonborn emperor". His grandson Titus Mede II inherited his throne, and now has helped outlaw Talos. And now a new dragonborn appears. Do you really think they should be unconnected with their predeccesors? Or worse yet, playing the part of a random subservient soldier (on either side). Or even funnier than that, "roleplaying" as some lumberjack or cook. This world should definitely end, if that's the case. The Thalmor should win, if the one person left on the planet to rectify the situation wants to be a nobody.
Maybe, but it just came into my memory that the civil war and the dragonborn are part of a prophecy: nothing political or religious in this, although a prophecy can be religion related.
When misrule takes its place at the eight corners of the world
When the Brass Tower walks and Time is reshaped
When the http://www.uesp.net/wiki/Lore:Tribunalhrice-blessed fail and the Red Tower trembles
When the Dragonborn Ruler loses his throne, and the White Tower falls
When the Snow Tower lies sundered, kingless, bleeding
The World-Eater wakes, and the Wheel turns upon the Last Dragonborn.
Anyway, the civil war has already begun when the dragonborn shows on the scene, so there's no real reason to bind the civil war quests to the dragonborn ones.