For specifying animations, you should work in the quaternion lie algebra, not in the group as you suggest. There you can represent 1620 degrees without any problem. Furthermore, in the quaternion Lie algebra (pure imaginary quaternions), and only in that space, you can take an arbitrary rotation key, multiply all 3 of its values with 10 and get 10 times that rotation without change in axis.
If you rotate around just one axis, the Lie algebra feels just like Euler angles .. in fact its exactly the same thing, but if you rotate around more than one .. it keeps working intuitively and usably - Euler angles absolutely do not.
These are also called the planar quaternions and are the even subalgebra of 2D PGA. I use and explain these (and their nD generalisations) in my SIBGRAPI 2021 talk on kinematics and dynamics in PGA.
Not the same concrete example, but one where I do find the Geometric Algebra version substantially more insightful, is the treatment of rigid body mechanics in the geometric algebra of the Euclidean group (R_{n,0,1}).
It has the dual quaternions as even subalgebra (in 3D), and unifies all linear and angular aspects. It leads to remarkable new insights, as removing the need for force-couples (pure angular acceleration is caused by pushing along a line at infinity), while pure linear acceleration is caused by forces along lines through the center of mass.
These geometric ideas are independent of dimension - forces, both angular and linear are always lines. The treatment of inertia becomes a duality map, and things like Steiners theorem are not needed at all.
On top of this, the separation of the metric that sets GA apart means that this formulation of rigid body dynamics works not only in flat Euclidean space, but unmodified in the Spherical and Hyperbolic geometries. (by a simple change of metric of the projective dimension).