The variance of the Gaussian you get isn't arbitrary and related to the variance of variables being combined. So unless you expect people picking folks out of a lineup to be mostly noise-free, a very narrow Gaussian would imply a violation of assumptions of CLT.
This Jewish law thing is sort of an informal law version of how frequentist hypothesis testing works: assume everything is fine (null) and see how surprised we are. If very surprised, reject assumption that everything is fine.
Thus our knowledge on people being noisy means the mean is illdefined instead of inaccurate.
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