Angular velocity of thrown object?

I was reading this and came across the statement

After releasing the knife, it will fly forward and continue to rotate around its center of gravity with the same angular velocity it had during the throwing movement.

However I'm not sure how to show that this would be true. Given the text diagram

Time 0  Pt A ------- %%CoM%%
Time 1  Pt A         %%COM%%

At time 0, the knife (or let's say rod) is rotating at $$\omega$$ (rad/sec) around Pt A. But does that mean the rod is rotating at $$\omega$$ around its own center of mass (CoM)? Pictorially, I convinced myself it's the case (see picture below). But I'm not sure how the equations would work out. At time 1 -- even if the rod's angular velocity is the same around the two points (Pt A / CoM), does the it stay the same after throwing?

I've tried to think about it from the conservation of angular momentum $$L$$ perspective. Say we define a system that only includes the rod. I was worried that the arm produces some external torque. But I think since we are only pulling tangentially, there might not be any external torque on the rod. In which case, the inertia of the rod (around CoM) does not change and the angular velocity (around CoM) does not change either. Then the quoted statement holds.

Am I overthinking this? I've just made myself more and more confused. Any clarity appreciated -- maybe this is the wrong approach entirely.

The equations work out because the net force (ignore gravity) is zero. And the net force describes the motion of the center of mass point only. You can see this clearly in this video depicting the motion of a rotating tennis racket.

$$\sum \boldsymbol{F} = m \boldsymbol{a}_{\rm com}$$

In addition, and regardless of the motion of the center of mass, the net torque through the center of mass describes the motion about the center of mass.

$$\sum \boldsymbol{\tau}_{\rm com} = \mathrm{I}_{\rm com} \boldsymbol{\alpha} + \ldots$$

In the absense of net force and torque, the solution is

\begin{aligned} \boldsymbol{a}_{\rm com} & = 0 & \boldsymbol{v}_{\rm com} & = \text{const.} \\ \boldsymbol{\alpha} & = 0 & \boldsymbol{\omega} & = \text{const.} \end{aligned}

Remember this rule: for a body to rotate about a point other than the center of mass a net force needs to be applied. The job of the force is to move the center of mass in a path that is not a straight line.

you may be over thinking it, the proof is, an object in motion tends to remain in motion. Once it is released it will continue in the same trajectory and rotation it had when released, until acted upon by an external force. usually, the external forces would be gravity, and air resistance, until impact.