So I think part of the answer is clearly saying what you want. I don't know if that holds for positions like Director /VP at large tech companies - these are always somewhat political - but it seems to hold up to pretty much right before that.
YMMV, of course. For all I know, I accidentally have the right magic keywords in my LinkedIn or something. (I don't know. I only update it every few years, so I certainly am not paying enough attention to tell :)
That’s probably safer for them. IME many recruiters doesn’t really have enough knowledge about the role they are trying to recruit for, so it would probably be very risky for them to try to find a ”rough diamond” since they are not really able to make such a judgement call with high enough chance of succeeding.
Define "recruiter".