(Spoilers Extended) The weirwood as a multi-verse shelter. "With great power comes great responsibility" for Bran, and Coldhand's comment.

Hear me out: if Bran could change the timelines, there would be a risk that he would erase himself from the timeline to begin with.

What if as part of the weirwood, Bran-TEC can survive any alteration to the timeline, even if it ended up erasing Brandon Stark, as long as the weirwood themselves still exist?

Imagine if Bran went back to the tourney at Harrenhal out of curiosity, and because he was so sure it was the Crannogman who was the mystery knight he ends up skinchanging Howland to play out the part himself, for fun, to be a knight for a day.

When he comes back to the "present", there's no Brandon to go back to. Bran is stuck in the weirwood, because he just altered the timeline; Meera isn't there, Jojen isn't there, Jon doesn't even exist, none of the Stark kids he knew exist.

Bran's accidental alteration of the timeline, due to how sure he was that it was the Crannogman who was the mystery knight when in fact it wasn't, has literally wiped out all of what we knew to follow the tourney, and who knows how things evolved from there. Catelyn may have married Brandon Stark, and other children were born of them, but not Bran. Maybe there's another Brandon, a first-born even, but Bran recognizes it's not himself.

Yet even if the Brandon Stark we know doesn't exist physically, the one who went in the weirwood still exists as part of it; he survived the timeline alteration thanks to being part of the weirwood net. Brandon Stark might not exist in this timeline, but the weirwood still do, and those north of the wall are far better protected from potential timeline alterations than those south of it as it's simply less likely that they would be wiped out all the way north.

So upon realizing this, Bran could go back and change the timeline again, to try and steer it to an outcome he desires. But if he goes back to before his intervention, hoping to let it play out instead, what if it now always flows in the direction he accidentally created? He would have to alter it again, trying to steer it back to what it may originally have been. And then, was the timeline where Howland wasn't the mystery knight even the original one?

In any case, Bran would have to go back and alter it again, hoping to bring it back to the timeline he knew, where he still existed. He might figure out the clues that preceded his previous intervention, and steer Lyanna to become the mystery knight once more.

So the weirwood is akin to a spaceship that could survive the big crunch, figuratively speaking for those into astronomy and physics.

But this also makes me think of Coldhands.

Meera's gloved hand tightened around the shaft of her frog spear. "Who sent you? Who is this three-eyed crow?"

"A friend. Dreamer, wizard, call him what you will. The last greenseer." The longhall's wooden door banged open. Outside, the night wind howled, bleak and black. The trees were full of ravens, screaming. Coldhands did not move.

"A monster," Bran said.

The ranger looked at Bran as if the rest of them did not exist. "Your monster, Brandon Stark."

People often say that Coldhands might be saying that he himself is Bran's monster, but the comment wasn't about himself, they are talking about who sent Coldhands; the three-eyed crow.

So Coldhands is saying that the three-eyed crow is Bran's "monster". Which makes it sound like the three-eyed crow is Bran's creation, or came to be as a result of Bran's action. It could also be that it implies that the three-eyed-crow is Bran's monster, as in the monster from Bran's perspective.

If Bran becomes the three-eyed crow and ends up in the situation I described above, Bran could definitely see the three-eyed crow as having been his monster, the one that put him in a terrible situation he now has to confront; Bran would now be responsible for steering the events of the world to bring it to a preferable outcome. In a way, this would perfectly tie in to the title of the last book, as the events which followed the false spring eventually led to Bran trying to bring about "A Dream of Spring"; it is a dream because it is a potential future, out of many, which Bran would now be seeking to bring about into reality.

Which might also imply that if he does manage to bring about his preferred outcome, he may well then decide to move on to burn all of the weirwood, to make sure that this power which he has ceases to exist. In the show, they ended with Bran looking for Drogon, and in the original outline Tyrion burns down Winterfell. I could see Bran seizing control of Drogon, masterless after the death of "Mhysa! Mhysa!", to burn all of the weirwood; to bring an end to the long night, which for as long as the weirwood and the power to constantly change the past exists, everyone was still figuratively asleep, existing only as part of a dream.