The rotational acceleration gained by a spinning ice skater can indeed be explained using the Coriolis force. Yes, there is no Coriolis pseudoforce in the inertial frame. However, the fact of the matter is that by contracting her arms, a sideways (tangential) force must somehow act on her body, for otherwise she cannot angularly accelerate. The question is, can this sideways force be explained from a fundamental viewpoint?
If we want to work directly in the inertial frame, we observe that
when the ice skater contracts her arms (along what she thinks are straight lines), they are actually traveling along curved paths. This requires a lateral force to be exerted on her arms, and if she exerts this lateral force herself (with her own body), then in the inertial frame, this must, by Newton's Third Law, induce a tangential counter-force on her body that rotationally accelerates her.
But now let's see what this implies in the rotational frame: Since a sideways force must be exerted to simply move the skater's arms in a straight line in this frame, this means there must be an opposing sideways pseudoforce that acts on any radially moving object in this frame. This pseudoforce cannot be the centrifugal force, since that only acts along the radial direction (and thus cannot push the skater's arms sideways). Thus, we conclude that there must be another pseudoforce apart from the centrifugal force in the rotating frame, and this must give any radially moving object a sideways push. This pseudoforce is precisely what we call the Coriolis force.
Thus, the rotational acceleration caused by the ice skater's contracting her arms can be legitimately attributed to the existence of a Coriolis pseudoforce in her rotating frame that she needs to counter to move her arms radially. Of course, since the same rotational acceleration can also be explained using conservation of angular momentum in the non-rotating frame, this shows that the Coriolis force and the conservation of angular momentum are intimately connected.