of the concept of parallel transport, which only works along geodesics, to transport along worldlines that might not be geodesics, i.e., accelerated worldlines
This seems to imply that you cannot make a parallel transport along a non-geodesic, which is not true. I believe what you mean to say is that the timelike tangent of a worldline is not parallel transported along a non-geodesic world line.
The problem with Fermi-Walker transport is that it requires at least three dimensions to make it clear about what ”non-rotating” means.
I would typically give the example of a circular path in R^3 (which is essentially the Euclidean version of your Rindler frame) and discuss that the tangent change is proportional to acceleration and in order to keep a vector orthogonal to the path orthogonal to the path you would need to change it accordingly proportional to the velocity.
Yes, it is the same in the Euclidean case, but you need the third direction to demonstrate how vectors orthogonal to both velocity and acceleration do not rotate, which was the point of referring to the frame field as somehow being non-rotating. I guess ”minimally rotating” would describe it sufficiently ...Not necessarily; the Rindler case in Minkowski spacetime can be reduced to two dimensions and will still show how the spacelike vector has to rotate to stay orthogonal to the timelike vector.