I like Chris Stapleton (but I think he's a little self-conscious in a way that works against his no-frills neo-outlaw persona). I didn't know the Piledrivers, though, and this song does exactly what you say it does. It's a real masterpiece of emotion. Simplicity is extremely difficult to get right.
John Stewart's "All the Brave Horses" gives me that frisson. He had worked on Robert Kennedy's presidential campaign and wrote this song in response to Kennedy's assassination, though it could just as easily be about our relation to nature and the wild. It's about as simple structurally as a song can be, but the performance packs a wallop.