Making use of the non-copyrightability of AI output to copy code in otherwise unauthorized ways does not seem like a straightforward or legally safe thing to do. That's especially the case because high profile proprietary software projects also make heavy use of AI, it doesn't seem likely the courts will support a legal precedent that strips those projects of copyright and allow anyone to use them for whatever.
I think what may happen in practice could be worse - basically if we can't tell whether some code is the work of a human, but the project accepts AI code, if there we forego the analysis of whether something was produced by a human, the entire project may be deemed public domain -- perhaps after a certain date (when LLM contributions were welcomed).
Beyond that, by integrating LLM code into those projects, the projects are signifying assent to their works to be consumed by LLMs for infringement of the whole work - not just the LLM produced portions - it is hard to be doctrinaire about adherence to the open source license when the maintainers themselves are violating it.
We may see a future where copyrights for works become more like trademarks - if you don't make any attempt to protect your work from piracy, you may simply lose the right to contest its theft.
Obviously, it is as you say - today the courts may smile upon a GPL project where a commercial vendor copied and released as their own without sharing alike - but if they instead say that they copied the work into their LLM and produced a copy without protections (as chardet has done), the courts might be less willing to afford the project copyright protections if the project itself was making use of the same copyright stripping technology to strip others' work to claim protections over copied work.
Besides which, "authored by Claude" seems like a pretty easy way to find public domain code, and as Malus presents, the only code that may ultimately be protected is closed source code - you can't copy it if you don't have the source.
The diversion of "people may try to pass of LLM code as their own" is a nice diversion, but ancillary to the existing situation where projects are incorporating public domain code as licensed. We can start there before we start worrying about fraud.