This is exactly why I hate time travel stories. It gets too confusing and nothing makes sense.
You didn't read the Harry Potter books, did you?
What I'm proposing is that Harry be sent back in time by GK to learn more about the Black Council, nothing else. Of course, then he ends up playing a role that he didn't know he would.
For example, Future GK helps send him back specifically to NOT change the past and only observe the BC shenanigans. And one of the first things he does is accidently bump into PG GK. PG GK realizes he's from the Future, and threatens to punish him for breaking a law. Future Harry manages to convince/prove to PG GK that he traveled with Future GK's help and consent. PG GK lets him go, and as he's departing, Future Harry makes an off-the-cuff comment about giving him a little more detail this time when he warns him about the black magic stuff. PG GK says, "um, I have no idea what you're talking about". And Future Harry says "yeah, you tell me there's dark magic being performed in town, and you have me look into it." To which PG GK says, "First I heard of it." Which confuses Harry, until he begins to realize that he sets that in motion. Which then has him worrying about what else he may have already set in motion the first time, but doesn't know what he has to do this time around. Queue internal dialogue rant about making things difficult for himself.
Thus Harry has to balance not changing anything other than what needs to be changed to match the first time around, while also trying to investigate and learn more about the BC.
Besides, using Knnn's theory there's no need to resort to time travel devices to make sense of the book.
No, time travel isn't needed to make it work. But time travel could be used to explain things in the book that went unexplained which his theory didn't touch on, and could also be the way that Harry finds out about the plot which Knnn spelled out.