Rotational Mechanics: Is Angular Acceleration Possible without any External Torque?

The definition of torque is not $\tau=Id\omega/dt$. We can't even define things like $I$ and $\omega$ for rotation that isn't rigid.

The definition of torque is $\tau=dL/dt$. So yes, it is possible to have an angular acceleration without an external torque. Your example shows correctly that this can happen.

so we can conclude that the man has angular acceleration without any external torque, which is an apparent contradiction of the terms, so how do we reconcile the case with the concept?

We reconcile it with the law of conservation of angular momentum.

The angular velocity of the skater increases when drawing in the arms in order to conserve angular momentum. The angular momentum of the skater will not change unless an external torque is applied to the object. So converse to your thinking, the change in angular velocity is due to no external torque being applied to the skater in order to conserve angular momentum.

Conservation of Energy:

The increase in angular velocity can also be explained by conservation of rotational kinetic energy. Ignoring friction there is no external force that can cause a change in the skaters rotational kinetic energy = 1/2 I$a^2$ where I is the rotational moment of internia of the skater and $a$ is the angular velocity of the skater. When the skater pulls his/her arms in it reduces the rotational moment of inertia I. In order to conserve kinetic energy the skater’s angular velocity $a$ must increase. Note however you can say that an internal force is what enabled the skater to pull in his/her arms.

Hope this helps.

When you pull your arms in you aren't pulling them directly towards the centre, because you're rotating as you're pulling them in. This is where the force comes from that actually makes you spin faster. You should definitely watch this video where he explains exactly this. Skip to 10m in if you're in a hurry but the whole video is well worth watching.

I know what you mean that "conservation of angular momentum" explanations can feel like they're hiding the details of the actual forces and torques going on. You can make a case that the conservation laws are actually more fundamental but either way both explanations are always possible and always give the same result.