I've been the sole admin of the LinkedIn group (me on SO and LinkedIn) since I proposed it in Stack Overflow on social networks (LinkedIn, Facebook, hi5, Orkut).
I'd welcome Jeff and co to the group - anyone who's a moderator on here would be welcome as an admin on the LinkedIn group. I'd welcome anyone with a decent rep from any of the SO sites as an admin - currently I have to manually accept all members, so sorry for any delays.
Better links is a good idea - lots of recruiters use LinkedIn, but they don't really know SO.
I think we would benefit from adding a LinkedIn 'application' (basically a profile plug-in) to display SO flair, but I haven't got round to looking at that in detail yet (I probably will next time I'm job hunting).
It should be possible using the LinkedIn API to post and update jobs from http://jobs.joelonsoftware.com to the group discussion board, but I'm not sure of the value - I spend more time here than on LinkedIn, so I'm more likely to see a relevant job advert on here than on there. I'd happily support someone else doing it if they felt that it would really help though.