Nah, its the collective that makes us what we are. Shanks had Bob, Joe and Ronnie, Bob had Joe and Ronnie and Roy, Kenny had Ronnie and Roy, Rafa had Pako, Klopp had Buvak and now Pep, plus all the others such as the medical staff. While the manager is a big part, I think even Klopp would agree that without the backroom staff, he couldn't do his job as well as he does. When Shanks quit, it was likely the solid backroom staff that allowed Bob to seamlessly move on and then Joe and Kenny found it the same.
In terms of the daily work, of course it's about the quality of the collective, all the departments, to give you what you don't have / know.
The quality of the manager (aside from his technical level) is in assembling the structure: picking his staff, keeping up with the field, even innovating, cutting loose staff when it's time, creating the right environment. These qualities are not democratic, nor transferable, they vary by the individual. This is what you lose when you lose a top manager. You lose that structure, that cycle will decay and end.
For example, from your list, Rafa and Klopp replaced Pako and Buvac and went on to title challenges, trophies. Those guys have done nothing since.
The following is caveated by the fact that I didn't live through the detail (started following us late Souness onwards) , it's off reports, sure I've done the research but it's still going to be second rate:The talk of the boot room, while nice, is still secondary to the decision making qualities that Shankly, Paisley and perhaps Kenny brought to the table, at their peaks. Guys like Souness, Evans, who were part of the same group, simply weren't good enough as managers, having had the same exposure. The environment is useful, to an extent, but it's always secondary to the decision making qualities and action that shape a reality.