I don't think giving Jon and his line the throne would solve anything (aside from the fact that I highly doubt either Jon or Dany will survive the series). I think the problem is more fundamental than that: As a decentralized feudal monarchy, the Seven Kingdoms are ungovernable. As a single political entity, it's too big, and the rule of law is too weak (the latter being true both in terms of the power of the crown, as well as how poorly enforced laws actually are at the local level). The most likely scenario, IMHO, in terms of real life politics, is actually that the Seven Kingdoms dissolves back into some approximation of its component parts.
If you want to retain the Seven Kingdoms as a unified political entity, I think a move toward a more collaborative governance model could maybe keep it together a bit better, say, a "small council of governors consisting of Jon (Targ), Sansa (Stark), Gendry (Baratheon), Sweetrobin (Arryn), Yara (Greyjoy), Tyrion (Lannister), Edmund (Riverlands), Tormund (Wildling), X (Tyrell), Y (Dorne), and probably Z (Dothraki), where X, Y, and Z are whomever makes the most sense from those Houses. Either way, I think the Iron Throne is ultimately an impediment to long-term stability.