Anecdotally I've had more recruiters reach out to me for potential jobs when I have the Open to work badge. Why not have it on all the time even if you are currently employed, especially if you are well performing. Would that not be an incentive for your current employer to boost pay/bonus to retain?
Probably not... tbh, unless I'm looking, I stay the hell away from LinkedIn generally. What got me even considering a few jobs ago was evaluating some of their eLearning courses that mgt had wanted people to make use of... Not the best idea imo. So, you're going to push your self-motivated employees to a site that centers around career connections and recruiters?
In the end, I don't think most managers care that much until it's too late. Not all, but enough.