You're thinking about REW and Audyssey the wrong way. They're not equivalent.
Audyssey adjusts processing to try to overcome problems it your physical set up (the room). It can only work from the logical standpoint that the speaker's performance can't be changed. It also assumes that if it achieves the preprogrammed target response. It works as a system only - the mic is not linear, but the software knows how to compensate.
In contrast, REW allows you the freedom to tease apart the performance of the speaker (or even speaker components) from setup/room or treatments. The cost is that you need a legit mic.
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