True. I'm always interested in what would have happened if Harold had won at Hastings, to see who would have succeeded him. I'm of the belief that if Harold wanted his sons to succeed him then he would have had to work hard for it. For one, I think, most non Godwinsons would have voted for Eadger and two, in the eyes of the Church most of his sons were bastards.
Though looking at what you said about the trait again, it might be that only the sons born after Harold became King would be eligible for the trait, which would make sense in my book. From what I gather, anyone of the House of Wessex should get the trait because of the dynasty member who is King bit, so Eadger's sons will probably get it.
I started a
scenario once based on the premise that Harold lived, trying to rein in the spiral of events so that they all made sense logically and didn't get too extravagant (as can happen so easily with CK).
What I found interesting though was that Harold's sons by Edith Swanneck were all considered legally legitimate by English common law, even though the church didn't consider
more danico marriages traditional. They weren't ever bastardized. King Svein of Denmark, Harold's first cousin, had scads of "illegitimate" children with common law wives and concubines, but that didn't stop a good half-dozen of them from succeeding him to the throne.
What I imagine would have happened is that the Godwine family would have absorbed the last vestiges of the Wessex dynasty. Marrying Godwine Haroldson to Margaret is all that would have been needed to secure the dynastic link. Whether Eadgar would have been elected down the line depends on how competent he proved to be as an adult. Since Edward the Confessor had treated him as a sort of pseudo-adopted son, he was still in the care and household of Harold's sister Queen Edith at Winchester. All of this means that Eadgar is sort of Harold's de facto nephew, even though he's not actually closely related by blood. I think Harold was too shrewd to simply murder him off -- his culpability would have been obvious and it would have damaged his reputation. He would either have neutralized him by making him appear weak, or he would have co-opted his legacy by treating him as another son himself. He could easily have married him to one of his daughters (Gytha or Gunnhild) to make a son-in-law and just another part of the family. Thus, even if Eadgar did get elected, any potential heirs he sired would still carry on the Godwine bloodline.
Then again, a lot would be predicated on how Harold's sons felt about Eadgar. I expect that there would be a bigger feud between the children of Harold's two wives after he died with all the earls divided between the two factions -- the children of Edith #1 supported by Harold's powerful brothers and the children of Edith #2 supported by Harold's powerful brothers-in-law. And of course Eadgar stuck in the middle. I really should finish that scenario some day. It'll be epic.
But anyway, to tie this all in to relevancy, I don't think the Aetheling trait itself should necessarily be transferred to non-dynastic members in the absence of a viable heir. It should simply be possible for a powerful noble to win the support of the Witan in the absence of a mature, competent heir (ie. Eadgar was too young and inexperienced to be an effective foil to the Vikings and Normans). The Aetheling trait shouldn't
guarantee election, it should just grant prestige and encourage people to respect your claims.