Mike Babcock is purely a puck possession guy. You score goals by having the puck, you defend by having the puck. A lot of controlled exits and entries, not a lot of dump out or dump in hockey.
I think that coaches have a major impact on game results in the long term through their systems. Leafs are the perfect example for that. With Carlyle/Horacek they were a league worst in terms of CF%. Bring in Babcock, and that number (with far less talent) jumps to 13th best. Now, that also led to a drop to last place in the standings, but that's because the main player driven half of the equation (SH% and SV%) were 30th and 25th ranked respectively.
Now I don't think that coaches are the only driver of CF%, but they're a big one. Under Babcock in 2014-2015, the Wings were the 3rd best CF% team in the league. The year before that 9th, and the year before that 5th. It's pretty undeniable that coaches can greatly influence CF%.
Another great example is Darryl Sutter. Before Sutter, the Kings CF% numbers were 10th the year before, and then 16th the year before. Since he took over, the CF% numbers have either been #1 or #2 overall. In in that span, they've won 2 cups.
Same thing with Q. Top 6 possession team every year but last year since he took over in 2008-2009.
And the team track records leading up to these coaching changes?
Before Q: Chicago missed the playoffs 9/10 seasons leading up to him being hired.
Since Q: Made the playoffs every year, 3 cups.
Before Sutter: LA missed the playoffs 6 straight seasons, then had 2 1st round exits. Won cup the first year he took over.
Now obviously there's players who play styles that are conducive to those coaching systems that can raise the possession numbers further, but the main framework is set in place by the coaches. You can have great possession players who go to a poor possession team and their numbers drop. And vice versa. Phaneuf is another example. In 2014-2015, Phaneuf had a 45.2% CF%. During his time on the Leafs, his CF% with Gardiner (main pairing) was 53.2%. Traded to a lesser coaching staff with a lessor system, and he was at 48.5% CF% with Ceci (main partner). Coaching styles and utilization makes coaches have a huge impact on player performance as well