> I just don't like this characterization
That's ok. It's a personal value judgement.
However, the fact remains that rotations can "emerge" just from the desire to do additions and multiplications on tuples to be able to do polynomials with them ... which is more directly tied to its historical path of discovery, to solve polynomial equations, starting with cubic.
>historical path of discovery, to solve polynomial equations, starting with cubic.
Even with polynomial equations that have complex roots, the idea of a rotation is baked in in solving them. Rotation+scaling with complex numbers is basically an arbitrary translation through the complex plane. So when you are faced with a*x*x + b*x + c = 0, where a b and c all lie on the real number line, and you are trying to basically get to 0, often you can't do it by having x on a number line, so you have to start with more dimentions and then rotate+scale so you end up at zero.
Its the same reason for negative numbers existing. When you have positive numbers only, and you define addition and subtraction, things like 5-6+10 become impossible to compute, even though all the values are positive. But when you introduce the space of negative numbers, even though they don't represent anything in reality, that operation becomes possible.