My doubt is simple as that. When I have a smooth, regular curve (that is, its curvature is never zero), can I just assume that it is parameterized by arc-lenght, without any loss of generality? If not, is there any counter example?
In a word, yes. If a curve is $C^k$ smooth with nonvanishing derivative, then its arclength parametrization is also $C^k$ smooth. Thus, from the theoretical point of view, nothing is lost in the change of parameter. And something is gained: the formulas for the TNB frame become simpler when the arclength parametrization is used.
However, from the practical point of view the arclength parametrization is usually messy: even for the simple cubic $y=x^3$ it involves an elliptic integral. Thus, for explicit computations one usually uses the simplest parametrization available, be it arclength or not.