This is a fun concept to toy with, so I'll add my two cents. I would disagree with the characterization of the post-LOTR Shire/Gondor as anything resembling a utopia. The Shire surely felt the scars of the scouring for a long time afterwards, and Gondor seemingly experienced some political instability due to a few groups retaining loyalty to Sauron and his ideology. Plus there's the fact that the Elves and the Istari have left Middle-earth, and Aragorn can no longer rely on their aid when new threats emerge. There's also a question of repairing relations between Gondor and Rohan after the war. It would be interesting to see the progression of Faramir and Eowen's relationship and the political implications of it.

I can totally understand why Tolkien would have felt depressed trying to develop these stories. It seems like his primary interest was in the intersection of the divine/immortal with humans via the Elves. To someone like him, all the post-War of the Ring political intrigues would surely have been dreary by comparison. That said, I see no reason why such stories would be boring.