The confusion here is that many very able physicists have picked up the idea that motion at constant proper acceleration is somehow motion at "constant 4-acceleration" and thus "constant 4-force", but this is simply not true.
For rectilinear motion at constant proper acceleration $a_0 = A^\mu A_\mu$, the invariant size of the 4-acceleration is fixed and so is the direction in space, but the direction in spacetime is not. The direction in spacetime is orthogonal to the 4-velocity and the 4-velocity is certainly changing, and so is the 4-acceleration.
So the problem here is really about clarity and precision of expression. Many people say "constant 4-force" or "constant 4-acceleration" when what they mean is "constant size of 4-force" or "constant size of 4-acceleration".
Compare it to another case: motion in a circle at constant speed in Newtonian physics. Would we say of such motion that the acceleration is constant? We probably would, but we are familiar enough with this case that we don't confuse ourselves: we know the acceleration vector is changing, but its size is constant. Would we say the velocity is constant? No we wouldn't. Would we say the force is constant? We might, or we might not. Strictly, as a vector quantity, it is not constant.
So now let's come back to special relativity. If someone says "calculate the case of motion under a constant 4-force" then really the careful student has little option but to take the statement at face value and try to solve for that case. The solution is difficult and such a force certainly is not pure, and has very little relevance to physics. If the question was "calculate the case of motion under a 4-force of fixed spatial direction and constant invariant size" then the student (whether careful or not) can breath a quick sigh of relief and get on and tackle that standard problem. But if the question said the first while intending to mean the second, then it was an ill-posed question.
(An example of this came up recently in an Oxford physics exam; we need to improve our procedures to catch such things.)