It may be unusual to use accelerating reference frames, but they work just fine in classical physics.

No they don't. From an accelerating reference frame, an object with no force on it will accelerate. You can only get it to work if you add a fictitious force.

I don't think it's accurate to call it incoherent for accelerating reference frames. If you try to alter the coordinate system so that something that was accelerating is at rest, and you try to predict what happens with the normal laws of physics, you'll get the wrong answer. But it never says you should get the right answer. There's symmetries in the laws of physics that cause them to be preserved by Lorentz transformations. Since a transformation can be found to make an arbitrary object be at rest at the origin and in a given orientation, it's often useful to use the transformation so that you can do the math with the object being at rest. Special relativity simply does not have such a symmetry to allow an accelerating object to be changed to an object at rest.

The fact that you can't use arbitrary "reference frames" doesn't mean that special relativity only works for a special case any more than using the (x,t) |-> (-t,x) transformation on Newtonian physics not working means that Newtonian physics only works in special cases and is incoherent.

The reason special relativity is a special case is that it only applies to flat spacetime, when no mass is involved.