Actually, I find most fanfic for that show leans away from intense emotional excavation. 'King of the Hill' has such a specific, deadpan tone that trying to graft high-drama interiority onto Hank or Dale often breaks the character's voice. The emotion comes through indirectly, in the spaces between the propane jokes.
What I see working better are quiet, domestic moments that the show hints at but never lingers on. A story about Hank quietly fixing something for Bobby, not saying much, but the act itself carrying the weight. Or Peggy's private frustration after a failed substitute teaching day, shown through her overly aggressive typing on her laptop. The fanfiction that resonates doesn't force them to articulate feelings they'd never articulate; it builds the scenario where the feeling is just... present, in the background, like the hum of the AC unit.
It's less about exploring hidden depths and more about respecting the subtlety of the original material. The best ones make you feel for the characters by keeping them utterly themselves.