Hypothetically, you could model your puck as a collection of $N$ atoms glued together with some kind of force, and solve the equations of motion (Newton's 2nd law) for the collection of atoms when you applied the external force to the puck.
At no point in the simulation would the simulator ever use the concept of angular momentum. But (if the simulation was good), if you measured the angular momentum of the solution, you would find it was conserved.
So, no, you don't need to directly use angular momentum to solve a physics problem. But whether or not you use it, it is conserved. And if you use it, you dramatically simplify many problems in rotational motion, to the point where they reduce to "impossible without doing a complicated computer simulation" to "doable with at most a few pages of math."
Additionally, regardless of solving specific problems, knowing angular momentum is conserved gives you a lot of insight. You can use conservation of angular momentum to debug the simulation, for example. Or, you can predict what will happen in a complicated scenario where you can't accurately simulate all of the details, such as predicting how galaxies will rotate as the mass falls in due to gravity. Without insight, we would just be randomly trying to simulate complicated systems, and would not make much progress in understanding Nature.
Finally, in more fundamental areas of physics, like quantum mechanics, angular momentum is absolutely essential for understanding what is going on. Spin, for example, is a form of intrinsic angular momentum. There is no deeper explanation of spin than to say "this is angular momentum that an electron has when it is at rest."
Therefore, while you can solve classical mechanics problems involving rotational motion or rotational stability without ever talking about angular momentum, to do so would be very foolish.