Let them creep all over my profile: so far the only downside is that I have a pile of messages to sort through and say "no, thanks" to.
As soon as I would even touch one word in my profile, a pile of recruiters would be triggered instantly. From experience I know that these recruiters have zero added value for me and often even their customer. On the contrary even, they have a tendency to try to fill any job with any qualification, because when they succeed, they hit a jackpot.
Job vacancies are present on job sites like 'Indeed'. It is very easy to set an appropriate filter and just start sending out your CV to companies.
I got my current job because a recruiter cold-called me about my Linked-In profile. It was a perfect match for a position she was trying to fill.
I wasn't looking, but it wasn't hard to persuade me to switch either.
The majority of job leads I got from maybe the start of LinkedIn to about 8 years ago were crap.
These days, most job leads are not crap, they just are not competitive with my current job. Much improved! Granted, I just updated a couple months ago my profile to say, "Not open to opportunities" and because my profile is recently updated, I get people contacting me daily - even on the weekend.
But... every other recruiting email I've gotten has been at best something I just wasn't interested in, while the vast majority of them are so poorly targeted, I'm embarrassed for the recruiter for clearly not having a clue how to evaluate if a candidate is a good fit in even the most basic ways. All promising job prospects I've had (whether they worked out or not) came through connections, or active work on my part to seek out positions that interested me.
(Obviously everyone's experience differs; I see you mention downthread that you once found a great match from a cold recruiter call.)
So... did that one life-changing job come to me because of a privacy-minefield site like LinkedIn? I'm not sure how that in-house recruiter found me: it was a cold email to an address that I hadn't used in years, and it was just dumb luck that I signed into it a week after it was sent, which seems at odds with the usual way to get in touch with someone you've found through a job-network site.
I'm at a point where I don't think I really need to maintain a LinkedIn profile in order to achieve my employment goals... but a part of me is too afraid it'll be useful (or even critical) to something in the future, so I haven't deleted it. Meh.