I understand your drift but I don’t think it worth thinking about as I can’t imagine that city would have ever employed Klopp, their project was built around eventually getting pep and it then transitioned into having pep.
Beyond creating a winning machine, in a wider sense they’re also keen to present themselves well. Would they have wanted a nasty piece of work like Kloppo pumping himself up and unashamedly boasting about every small victory and spitting hate & blaming grass etc. for his failures while gurning at the cameras with his comedy teeth . . I doubt it.
It can’t be said that Kloppo doesn’t have a notable effect though. He’s galvanised football inasmuch as while most football fans (not dippers and not main rivals) were once disinterested or ambivalent about liverpool (in a meh so what? kind of way) nowadays you rarely meet a football fan at any level who doesn’t take pleasure in seeing them lose and knowing that horrible man will be gurning away while snarling out his ridiculous excuses with spittle dripping off his chin.
As an example, I support a team in lower league football and, when PL scores are given out over the tannoy, if the dippers are losing there’s always a cheer from the fans of both sides involved.
It takes a special kind of tw@ to generate that kind of response from such a wide cross section and, even though it’s an achievement of sorts it’s not exactly the kind of representation that a decent club would want eh?