I think part of the problem is that in the CoC context, the slighted party now looks at your username, cross references it against reddit and twitter, searches your posts for something else to take umbrage at, and makes the case that you should be excluded from the community. The result is a very real chilling effect on speech on all public platforms because now you're never allowed to make a comment that might be taken out of context to offend now or at any time in the future.
This is the key to the whole thing and I'm glad someone finally said it. Open-source projects aren't incapable of getting rid of contributors who cause problems to the detriment of project productivity without having a formalized Code of Conduct. Why add something to a project that causes more problems than it solves? CoC proponents have good intentions but fail to see the chilling effect potential, even as it plays itself out in real time, right on cue (Ts'o).
I agree with this worry and think a CoC should make clear that it covers conduct within the project only.
That said, this chilling effect seems to come more from companies and public institutions in the UK/US than CoCs in the OpenSource community. I agree that this is something to clearly position against.
And I believe a well written CoC can help here by making this explicit: For example: By refering to behaviour in an official capacity. There is some subtlty with OpenSource projects and less formalized roles here. So that's a discussion worth having.