EDIT1:
A 3d curve passes through the origin, the $(T,N,B)$ are tangent, normal and binormal vectors respectively. In the (T,N) spanned plane of osculation can we parametrize a helix of osculation (with given curvature $\kappa$ and torsion $\tau$) as
$$ (x,y,z)= \left(\frac{\kappa}{\kappa^2+\tau^2}\sin t,\, \frac{\tau}{\kappa^2+\tau^2}(1-\cos t),\,\frac{t \tau }{2\pi(\kappa^2+\tau^2)}\right)\,$$
(where $t$ is rotation in the projected osculation plane around vector B ) ? If not, how is it correctly parametrized? Thanks in advance.
there are some papers (that also i contributed Öztürk, 2021(Nonlinear transformations)) in the literature which investigated the osculating helix. See (Bhat, 2018) for basics. I have to say that your thought a litte false, firstly you have to guarantee the radius and pitch functions r=k/(k^2+t^2), p=t/(k^2+t^2), where k and t are curvature and torsion of initial curve respectively. There is no need to be axis of the curve is a line. It is defined as axial curve of osculating helix and generally it is not line. Moreover, the base curve not need to be circular. But we can say that our curve has the form a(t)=(rf(t),rg(t),(p/2pi)h(t)), where f,g and h are differentiable functions of t. I hope this will be useful for your further study.