I took markjo's advice and did the math.

According to caltech:How fast does the Moon travel around Earth?

The Moon orbits Earth at a speed of 2,288 miles per hour (3,683 kilometers per hour). During this time it travels a distance of 1,423,000 miles (2,290,000 kilometers).

Moon orbits earth at speed of 2,288 mph

Circumference of Moon's orbit: 1,423,000 miles

Circumference of Moon's orbit / 360 degrees = 3952.77 miles per degree

2,288 (Speed of Moon in mph) * 12 = 47433.24 miles around the earth over 12 hours

47433.24 miles / 3952.77 miles per degree = 12 degrees around the earth in 12 hours

So the Moon is slower than the Earth's rotation. After 12 hours the moon would make an arc 12 degrees around the Earth, while after 12 hours a point on the Earth would turn 180 degrees.

Surely, I must be doing something wrong?

In which case, if the Moon is slower than a point on the Earth's surface, then the path of the eclipse should travel from East to West, as if the Moon were static over the earth while the earth turned beneath it.