A coach definitely needs to know the situation and pick their spots.
Hindsight 20/20 from my experience, a coach should not run WITH his runners.
However, there are probably more positives than drawbacks if a coach runs at the same time as his or her runners (this is different than running with them).
A coach can be a buzzkill or a coach can be undermining his authority by running among teenagers. Or seem like a douche by doing his workouts and competing with kids.
That said, the most important aspect of coaching is to teach running and to hold a high school athlete accountable. Our team used to either run multiple reps over our xc course or certain designated road routes. When I ran and came across runners at various times over the course, I could check their splits and make sure their pace was right (plus a lot of them did not wear stopwatches). I also found that certain less motivated kids would walk the course or route if they were not being watched. By running at the same time, you either could hold them to account or get them off the team so as not to poison the other runners.
If you have an experienced team with quality runners, they will often hold other runners accountable. But even then, some kids are hesitant to snitch on others. However, I think if you have an inexperienced team or trying to rehab a program that had previously crap coaching, you might need to be more hands on.
I also think that a coach that runs has a touch more credibility at times. Running is often perceived as a form of punishment in other sports. Screw up and go run a lap. Kids don't automatically see it as a sport unto its own and actually being fun. When a coach runs, it starts becoming something more than a person in authority just barking orders at kids.
I would also add, we ran in a public park with a zoo with plenty of families around. Just for disciplinary reasons, it helped being nearby running at the same time just in case kids weren't behaving and were dropping f bombs. It would be an administrative headache if a local called the school or park and complain about kids cursing. They do represent the school and poor behavior reflects poorly on the school and coaches.