Most answers I have seen to the question of whether AI-generated content is copyrightable cite law in the US and other countries limiting copyright to human-created content. However all the answers I have seen fail to address the question of how a government copyright office and then potentially later a rights holder, copyright enforcement company, or court would be able to prove it was AI-generated.
AI platforms are often vague about retention and/or use of user input and AI output, and it is not clear to me how accessible this data would be to government copyright offices, courts, and copyright enforcement companies for purposes of determining whether a particular lyric were AI generated.
In some cases, already copyrighted content may appear verbatim in AI output, and to the extent this can be easy recognized as an existing work this would make the AI authorship question moot. However most of the time this will not be the case, especially where AI only enhances a user's original work rather than composing something from scratch or with minimal prompts.