The answer is, they didn't end up on that side. Copyright infringement does not involve merely using a copyrighted work. Copyright infringement involves copying part of a work's copyrightable expression into another Thing (for lack of a better word).
In the US, if no part of the Thing is substantially similar [1] to any part of the original work's expression, then the Thing does not infringe on the original work's copyright. The EFF's "side" is that in such cases, there is no categorical obligation to add/include copyright management information (CMI) of the original work to the Thing, and it makes no legal sense to argue that the CMI was "removed" from the Thing. Not every LLM output contains expression substantially similar to any particular copyrightable expression in the training set. Whether LLM output infringes copyright can be different on a case-by-case basis.
> "it is ok to strip out my copyright string from my code"?
No, if part of the output (whether human output or LLM output) is substantially similar to (even an obfuscated or mangled version of) the copyrightable expression in your code.