I think that's an over-reaction. I am in a similar situation (I don't want certain people to know where I work), but I am OK with keeping the old CV, which doesn't mention the place where I work now, still online.
I have also asked the head of HR to make absolutely sure that my name is not on their public "our team" page. As a side effect, I always giggle during the mandatory corporate security training that tells how to deal with targeted attacks and demonstrates example phishing emails - so far, I received zero, if we don't count tests by a pentest agency.
Yeah, today, if I needed to keep a lower profile about a role, I'd still want to be on LinkedIn (if only to be reachable by contacts, now that the Web is so centralized), so I'd do something like that.
I have also asked the head of HR to make absolutely sure that my name is not on their public "our team" page. As a side effect, I always giggle during the mandatory corporate security training that tells how to deal with targeted attacks and demonstrates example phishing emails - so far, I received zero, if we don't count tests by a pentest agency.