angular momentum - conceptual doubt in method to find moment of inertia about an axis

I asked this question before about whether I can take a component of angular velocity along another axis and say that the body spins about that axis with that component.

Now I have another doubt:

Consider a rigid body having an inertia $I_0$ and angular velocity $\omega_0$ about some axis. So according to the answer to my question above, I can say that the object has an angular velocity $$\omega_0\cos\theta$$ about an axis inclined at $\theta$. And I can also say that the angular momentum about that axis will be $$I_0\omega_0\cos\theta$$ by taking the component of the angular momentum about the original axis, $I_0\omega_0$ along the axis at $\theta$.

So why can't I say that the inertia about that axis will be $$I = \frac L{\omega}=\frac{I_0\omega_0\cos\theta}{\omega_0\cos\theta} = I_0$$

Where is the problem in this?