I visited some other QA threads about this topic, and I don't understand why people think it's mysterious that the bucket knows about its rotation.
If a non-rotating bucket is all there is in the universe, then, initially, all the parts of the bucket are at rest wrt to each other.
But if we want to rotate that bucket with an angular velocity $\omega$, then we basically require the different parts of it to have relative acceleration wrt each other. Because if we divide the bottom of the bucket into many concentric rings, then each ring would've an acceleration $\omega^2 r$ towards the center, depending on the radius $r$ of ring. This means that the rings have relative acceleration wrt to each other. Laws of physics would take different forms for people standing on different rings. Hence, a rotating bucket is a collection of non-inertial frames having relative acceleration.
But non-inertial frames are supposed to detect acceleration in Newtonian physics. So what am I missing?