Leadbelly is excellent, as is basically most of the early blues. Kansas Joe’s “When the Levee Breaks”, a lot of Robert Johnson, Little Hat Jones’ “Bye Bye Baby”, Mississippi John Hurt’s “Candy Man”, Etta Baker’s “One Dime Blues”, Elizabeth Cotten’s “Vastapol”, etc etc. Just guitars and a voice - not much too simpler.
Also, classic folk is great for this. There is not much complexity behind Woodie et al, because they were writing songs for the people. They wanted everyone to listen, and to participate, so it all feels so genuine and beautiful and simple.
For more modern stuff, there is too much to name. Most of Jolie Holland fits the bill, but that is because she is pretty much blues. There is also Neutral Milk Hotel, which has either complex or “simple” songs, but all are beautiful. Simple and beautiful would probably go to “A Baby for Pree.” Modest Mouse has “Talking Shit About a Pretty Sunset”, which is simple in context, but just barely on its own.
I also have this live track of Ben Kweller’s “Walk On Me”, and it fits perfectly, as well.
They Might Be Giants have “When Tornadoes Take Over the World” and “Where Do They Make Balloons?”
Simple and beautiful, but for it’s coarse, raw feel has to go to Beat Happening for “Jamboree.” In fact, everything they do is simple and beautiful, but in a completely different way.
Also, Sparklehorse almost always release(s/d) simple, beautiful stuff - unless they felt like rawking out. They have some so-simple-it-hurts songs on most of their work, and it all sounds beautiful.
However, out of all these, the most basic, beautiful song I can think of is, surprisingly, a Bright Eyes one. “Lua” is just him singing slowly, playing just as slow, but it seems so spontaneous and on the verge of collapse that you can’t tell what to make of it. You never know when he’ll end, and you’re not sure if you want him too. Good stuff.