I've since come to the conclusion that when you mix simple tools with a bit of brainpower, you can accomplish far more than people would ever believe.
How does he get the pebble underneath a large block initially?
How will get the horizontal stones across the top lifted and pivoted into place?
I imagine once he'd gotten the supports placed, he could just raise the lintel between them with the machine, perpendicular to its final resting position, then rotate it into place.
To get an initial fulcrum under a stone, it looks like he built a wrench-shaped wood frame, essentially attaching a long lever handle to the stone. One can be seen in the second picture on this page:
At 2:08, they show the counterweights and one of the pivots he used for moving the barn. (It's a "pole barn", so it has no concrete foundation. His webpage says they had to add about 50% of its weight again in reinforcement so that it would hold its shape when moved this way.)
He used a pivot sitting on top of wooden shoring boards, I can imagine the process for any weight would be similar.