Since you apparently quit the discussion just at the point where the crucial gap in your argument had been identified, let me summarize the findings of our extended discussion:
Our CNOT gate discussion started with your claim that it is a measurement device:
Our discussion revealed that if each target line is initialized with a definite zero state
, CNOT gates can be used to construct an ancilla for a sequence of quasi-measurements, such that the reduced density matrix on the output control line is decohered, i.e., diagonal. Therefore different observers see the same result, conditioned on a particular measurement result for one observer.
But this doesn't hold for CNOT gates that are differently prepared. This shows that the CNOT gate by itself is not a measurement device, but only the dissipative system that consists of the CNOT gate together with another gate
that prepares the target line in a definite zero state. The latter requires already a definite outcome of a measurement, and hence must be itself a measurement device.
Indeed, in quantum information theory, one has specific measurement gates that perform a binary projective measurement and produce a definite outcome. These gates exist as real devices, and are necessary for any quantum information technology.
Thus while CNOT gates explain the working of decoherence in a very elegant and simple way, they - like decoherence itself - do not explain the working of measurement gates (or any other measurement devices).