I think the sport in which coaching makes the most difference is American football. There, the game not only has a paper-rock-scissors dynamic, in which good gameplanning makes a difference, but the coach literally calls every play for his team. That's what happens less in soccer where after a certain point, players just play.I think it is completely the opposite in soccer. More than in other sports the two jobs should be intimately connected, as system fit is absolutely crucial in putting together a winning side.
In soccer, there are many ways a coach makes a difference. What you say about it's not necessary. While there are coaches who have their mind set on the way their team should play, for many coaches the mentality is how to squeeze the most out of a particular group without a preconceived notion on what formation or mentality a team should adopt. This seems the more correct approach to me.
I should also point out that there has been research done examining team performance before and after mid-season coaching firings and has found no difference in team results before or after. That's the case with Klopp last year as wellas well. I don't think that he's a bad coach, but if coaching made a difference, you would expect for the coach to lead to an increase in points per game while working with the same personnel as his predecessor. I also think that trying to explain Liverpool's variance in performance during the Rodgers-Klopp years is better explained by the changes in player personnel than trying to argue that Rodgers was great in 2014, then started sucking, then Klopp took over, but there was no change because he needed time to teach his system and then magically things improved after he did so during the summer - where coincidentally there was also roster turnover.
The funny thing about soccer is that the vast majority of coaches leave their teams as failures regardless of how many good seasons they have put in. Klopp had a few great seasons with Dortmund and when Dortmund had a crappy season which was likely just caused by bad luck, he was fired. All the same, all the coaches who are departing as failures tend to be welcomed as saviors replacing the current failing coach who fans can't stand. And that story keeps repeating itself without fans seemingly able to get a whiff of it.