I think the best way to start may be to ignore LinkedIn. I agree with everything you say about LinkedIn. It's the playground for sales reps, recruiters and HR people. And in that it does play a valuable role. Yet buried way back in time was this idea that you might connect to someone new through an existing mutual connection. There was trust based on that connection that made it a "warm" introduction. That concept is all but lost today.

I think you convince people to join specifically because they believe the only folks contacting them via this service will be trusted, warm and valuable. They won't get a lot of unsolicited requests from people who don't offer a mutually valuable relationship.