This is not an either - or situation. One should make a distinction between the physics of the effect, in this case the accelerating detector coupled to a quantum field registering the thermal spectrum of excitations of that field, and the conceptual framework used to explain the effect. Both GR and SR are such frameworks and each could be used to describe this effect.
Since the effect does not depend on the spacetime curvature, one does not need to use GR for the description, it is perfectly possible to describe the behaviour of accelerating detector using only notions of QFT in Minkowski spacetime (i.e. the SR theory). At the same time using the set of tools from general relativity (more specifically, from QFT in curved spacetime) also allows one to describe the very same physical situation. Moreover, there are multiple GR techniques that could be used to achieve quantitative description of the Unruh effect (Bogoliubov transformation method, WKB/tunneling method, anomaly cancellation method, etc.) each one offering new insights into the situation. The existence of such dual descriptions (SR and GR) is an important test of the formalisms employed and of our intuition built upon these sets of tools.