I just want to offer my perspective here, having been coached by two very different coaches in high school but both ran with the team at least on a semi-regular basis. My first coach ran with the team probably twice a week or so from what I can remember, very possibly less. Most of the time he didn't leave with a group, it was mostly bouncing around checking up with people. Occasionally in summer practices he would run tempos with us, but during the year he would observe, take splits, and adjust the workout if you were going too hard. He hadn't competed for years so any workouts or runs he did were just for the love of the sport. He was all business for the most part -- it was obvious that he wanted to be a coach, not a friend. He still had personal relationships with the guys on the team but he definitely wasn't "one of the guys." We all had a lot of respect for him and it was great seeing his love for the sport, and knowing that he knew how difficult the workouts were. On the flip side some people, in particular some of the girls, didn't like him because he didn't coddle their fragile mentalities.
The next coach was "that guy" that people earlier referred to. He ran with the team most days (girls as often as guys, which was strange) and tried to be "one of the guys" for the most part. He was still racing and did probably half of the workouts with the team for his own fitness. He would leave with a group and do the entire run with them. I think he definitely crossed a line being too personal in some instances. He was extremely personal with the girls team in particular (not in a sexual way) but his neediness to have the girls team like him, and in particular the freshman, was extremely inappropriate. I think a lot of the team lost respect for him every time he used practice time to rehearse his comedy routine with the girls.
Generally, I think a young-middle aged coach should run with the team. However, I think there is a strong correlation between running with the team and being "that guy." Knowing your coach is a serious runner definitely earns trust, and seeing it with your own eyes is even better. The goal of the coach just needs to be in the right place. A coach's job is to make his team faster and anything he does should be done with that purpose in mind. Coaches should not want to be buddies with high schoolers and it's sad if any 30-something guy would want to. If you want to develop a personal relationship with your athletes it should be to improve your team, not for your own personal enjoyment. Hope this input helps.