If I had to build something like this in 1000 BC, I'd use a cart with wheels that did not turn at all and put a heavy statue on top. Then, whenever it came time to change direction, I'd get a few guys to hoist the statue up, while someone else rotated the cart independently. (And there are easily mechanical ways to make this process simpler.) You could recalibrate using the sun or stars.
A differential gear would automate all of this, but would be unmaintainable, expensive, would barely work for more than a few hundred yards, and all this in an era where labor is slave-cheap.
EDIT:
Another way would be to use multiple carts, only move one at a time, and calibrate it from the others, survey style. Cumulative error should be far smaller this way, and you would have fewer terrain problems.
It would be a massive annoyance, but would still work better than a differential gear made with the materials science of the time. I personally think that the story is fantasy.
You can still rotate the entire chariot. There’s a vertical axis of rotation through each wheel. Think about how you can rotate a wheelbarrow by rotating the entire thing at once. The tire doesn’t spin, but you’re still rotating.
Perhaps he meant a fixed axle, so both wheels have to turn at the exact same rate. Still doesn’t make much sense though, it would be incredibly inefficient to have to lift up the statue every time the road changes direction ever so slightly.
A differential gear would automate all of this, but would be unmaintainable, expensive, would barely work for more than a few hundred yards, and all this in an era where labor is slave-cheap.
EDIT:
Another way would be to use multiple carts, only move one at a time, and calibrate it from the others, survey style. Cumulative error should be far smaller this way, and you would have fewer terrain problems.