What is the best way to introduce the notion of angular momentum to a class without making it appear an unnecessary and artificial construction?
I suggest a way: bring a toy of a gyroscope form, put it on a table, and give it a brief torque. Although you don't act anymore on the toy, it continues to rotate. Ask your students WHY does it happen.
I assume that they learned about the conservation of LINEAR momentum. So, we have an analogy: a body in linear movement keeps moving as long as no force acts on it, and that because of the law of linear momentum conservation; analogously, we have a law for ANGULAR momentum conservation. This law keeps an object in rotation.
Now, you have to define what is angular momentum. It goes also by analogy with the linear momentum. The latter is defined as
$$p = mv$$
where $m$ is the mass of the object, and $v$ is its linear velocity. In the same way,
$$L = I\omega,$$
where $I$ is named momentum of inertia, and $\omega$ is the CIRCULAR velocity, the angle by which the toy rotates in a unit time.
After that you can explain (on a circular object) that it was discovered that
$$I = \sum M_i r_i^2,$$
where $m_i$ are the material points in the toy, and intuitively, the bigger is the toy radius, the bigger is the linear momentum, i.e. it will be more difficult to stop from rotating a toy of big radius, than one of small radius that rotates with the same $\omega$.
Later on, to explain for instance rotation of planets, consider what happens when the toy is hollowed around the axis, and next, consider that just one of the points $M_i$ of the toy.
Just two cents: I assume you already introduced Newton's laws, you can say that is something like "When viewed in an inertial reference frame, an object either remains at rest or continues to move at a constant velocity, unless acted upon by an external force" yu can explain that from the other two Newtons laws it can be shown that the first law naturally applies also to objects that move on circular motion, not only in a stright line. Then explain that this amount of rotation implies that if the objects contract due to internal forces (no external influences), the amount of circular motion is still constant, but the angular speed needs to augment because now it has to cover a smaller circumference with the same amount of rotation. Show the example of the ballerina that speeds up as she contracts her arms.