What's so bad about a "generic" resume? I assume this means one that just honestly describes your experience rather than tailoring it to the job applied for to make it seem you're a better fit than you really are. It's up to the person (hopefully) reading your resume to decide whether you're enough of a potential fit to take to the next step (technical screening call?).

A cover layer is there to show how you fit the role. This cannot be done with a genetic cover layer

Telling facts about yourself is done in the CV, not cover layer

Sure, but the parent was talking about customized resumes, not cover letters. The cover letter of course needs to be customized, else it serves no purpose.