I don't think this would add relevant information to Stack Overflow as a site. SO is a site for answering and finding the answers to programming related questions. Whether the asker or answerer is currently searching for a job has no relevance, and should not be displayed in any way with the their information on the question page. Any person can choose to add a link to the public CV to their profile, because your SO profile is your sandbox to put out there any additional information you want people to have. Furthermore, I'm not convinced employers would or do search SO for people to hire, I would expect they will use careers, which has a search system tailored to helping them find the potential employees they might want to hire.
Linking to the profile could make it discoverable by search engines. That's intentionally not the case so you can keep your "public" CV from being found accidentally by someone you don't want to know about it -- such as a recruiter. I do have my CV linked in my profile, though I've added a disclaimer requesting no recruiters use it to market me -- at least without my consent.