Teams pay the coaches to do their job to the fullest of their abilities it would be a waste not to have the coaches helping the players during the pick ban phase.
It's been going on in the LCK for years since 2013 with coaches helping the players out with pick ban and they've turned out pretty good. NA just started it and it's actually been helping the region produce and look good since, so I personally don't want the coaches to go away from the pick bans.
You watch solo Q streams and the pick bans are all over the place that's what the LCS would be just sporadic and all over the place but instead teams have coaches and their job is to look up pick bans, objectives, planning 24/7.
A coach can tell their players before game "ok GL and HF etc" and tells them about the game plan etc in the back but once on stage I'm pretty sure some if not all get goosebumps and are at least a little fearful before games and before the champion select, so anxiety kicks in but if a coach is on stage with them helping them with champion select it helps eliminate some of the anxiety and helps alleviate the pressure.
A coach can also joke with the players and that too helps. Think of as a teacher that is cool and jokes around in class and isn't just strict or 1 of the classmates so the passion goes into the school work from the students knowing that they can have fun instead of just performing.
Another thing is that with a coach helping with the pick ban and that team loses it's much easier for the bigwig that owns that team to fire and put all of the blame on the coach instead of their talent as it's easier to get another coach than it is to get a potential superstar. Example would be Locodoco from TSM.