Why do Rogowski coils work for measuring current?
A Rogowski coil is not a current transformer.

The current in the main conductor generates a magnetic field round itself. The Rogowski coil samples the field, generating a voltage as the field changes. It is weakly coupled to the main conductor. As little, ideally no, current flows in the Rogowski coil, there is no effect back on the main conductor.

Any loops couple. The sampling coil would generate a voltage whether it formed a closed loop around the main conductor or not. However, that's not terribly useful, as the coupling, and hence the generated voltage, would change if the conductors changed position. The sampling coil would also respond to any current changes anywhere in the vicinity.

The reason the Rogowski coil takes the form of a uniform toroidal winding is because due to the symmetric situation, it couples to all the current flowing within the toroid, regardless of the position of the conductor. There is zero coupling to any current that does not pass through the hole. An incomplete circle, or a non-uniform winding, would not have these two desirable properties.