It will take the new coach beyond a season to prove he was better than klop, the achievements of klop in Liverpool is just so much that he can't just beat the recordd in a season and expect to be seen as been better than klop was all the years he was at Liverpool and even gave the team a very good face lift across Europe,in subsequent seasons if he's able to consistently get the team to good points beyond that point which they have been, we could say that chances are that he would want to supercede klops records.
You are right, a new coach will definitely need time to show they are worthy of being a coach, especially if he replaces Jurgen Klopp who has been at Liverpool for years. That is definitely something difficult, because the philosophy that Jurgen Klopp has built may change and of course it will require a transition, especially if most of the players are Jurgen Klopp's legacy.
Arne Slot has a task that is not easy, he bears the burden of everyone in order to bring Liverpool to play well, at least he must be the same or even better than Jurgen Klopp's coaching era. I hope Liverpool will also give him a decent opportunity, because he has to work under the shadow of the achievements achieved by Jurgen Klopp during his time at Liverpool. I hope he can also relax or not feel burdened by the expectations on his shoulders. Because any pressure cannot be controlled well, it will also have a negative impact on Arne Slot's work.