No, it's not Brynden. It’s Bran.
FIRST -> WHY THE CROW IS NOT BRYNDEN
the short answer is "he literally doesn't even know who the crow is." but let's dig into that.
"Are you the three-eyed crow?" Bran heard himself say. A three-eyed crow should have three eyes. He has only one, and that one red. Bran could feel the eye staring at him, shining like a pool of blood in the torchlight. Where his other eye should have been, a thin white root grew from an empty socket, down his cheek, and into his neck.
"A … crow?" The pale lord's voice was dry. His lips moved slowly, as if they had forgotten how to form words. "Once, aye. Black of garb and black of blood." The clothes he wore were rotten and faded, spotted with moss and eaten through with worms, but once they had been black. "I have been many things, Bran. Now I am as you see me, and now you will understand why I could not come to you … except in dreams."
The last greenseer, the singers called him, but in Bran's dreams he was still a three-eyed crow. When Meera Reed had asked him his true name, he made a ghastly sound that might have been a chuckle. "I wore many names when I was quick, but even I once had a mother, and the name she gave me at her breast was Brynden."
He does not know what Bran is referring to even a little bit when Bran refers to him as the three eyed crow. He does use the "flying" metaphor but never once does he make any sort of reference to the three eyed crow - and nor do the singers. Also notable here that Brynden's imagery is often similar but slightly to the left of of the three eyed raven - Brynden has "a thousand eyes and one" instead of three eyes, and he is from raventree hall and associated with ravens more often than crows.
"but visenyas hill" you say, "greenseers don't know what form they appear in when they visit other people, so maybe he just didn't know he appeared as a crow to bran?" excellent point! CHART!
These ^^ are all the beings that appear in greendreams. There's a few interesting things here - firstly, that there seems to be two weirwood trees communicating through dreams, and the second is that the three eyed crow is significantly more active than any of the others. The Three Eyed Crow talks to Bran, teaches Bran, even makes fun of Bran a bit in a jokey manner. The weirwood Bran sees only ever says Bran's name...and the thing is that that adds up with what Brynden has said-
Now I am as you see me, and now you will understand why I could not come to you … except in dreams. I have watched you for a long time, watched you with a thousand eyes and one. I saw your birth, and that of your lord father before you. I saw your first step, heard your first word, was part of your first dream. I was watching when you fell.
WATCHING. And if you look at the chart, who is just watching? It's the old weirwood with the twisted mouth. The crow isn't just "watching" he is actively teaching Bran - often painfully. Brynden doesn't acknowledge that aspect at all. And again - he refers to himself as having "a thousand eyes and one" watching and not with his third eye.
Brynden never refers to his third eye and does not refer to himself as a crow
Brynden says he "watches" Bran, which the weirwood with the twisted mouth does, while the crow is significantly more active in Bran's dreams
The old twisted weirwood does not acknowledge the presence of the crow while Bran is dreaming either
Brynden's mother's sigil is....a weirwood with ravens.
SECOND -> NOW WHY IS JAIME IN IT?
Simply put, I included him because he is a constant spector in Bran's dreams. He's also a spector that the crow is aware of! The crow directly tells Bran to "put that dream away" and Bran responds by suppressing the memory of the golden man. But he can't hide from what he knows because later on, after a full book of pretending like he doesn't know who pushed him, Bran hears them talking about Jaime being Joffrey's father, and it immediately resparks his nightmares.
What's interesting to me is that this incident of falling, of being pushed, this forbidden knowledge, it's so tied to Bran's magic. Falling is what jump starts his magic. The mere memory of Jaime restarts it again, before he once again pushes it down. And Jojen's dream of the winged wolf imo ties into this a bit - Jojen mentions that bran still won't fly even after his third eye has opened. Coincidentally, after Bran's third eye opens, he suppresses Jaime successfully. I think part of Bran's problem with connecting to his magic is because he won't name the golden man, and he won't acknowledge his grief over what happened to him.
Also notable is that the same book Jojen says Bran refuses to fly is when Jaime has his first funky magical dream. Most of it is about Jaime himself but there's that little section where Brienne asks what it is they are chasing - she asks if it's a cave lion (haha, Lannister), a direwolf (hmmm), or some bear (bc the pit), and Jaime says it's "doom. only doom." The direwolf being mentioned in his dream is important to me - we know Jaime loves to dissociate and suppress his feelings, and I think the direwolf appearing speaks to his own feelings on what he's done to the Starks he's come into contact with. I think it's also interesting that the next time we see Bran, he is being told that he should not fear the darkness, but the darkness is what Jaime fears in his dream. For both of them, what they fear is the truth being brought into the light. The darkness only brings doom but the darkness is familiar, and so much less terrifying than the light of day.
Is this relevant to who the three eyed crow is? Well, again, I think it's relevant that the three eyed crow tells Bran to forget about Jaime and "put him away" for now. The Three Eyed crow just seems a bit more engaged, a bit more aware of what it is Bran is struggling with.
NOW FOR THE REAL MEAT AND POTATOES -> WHY THE THREE EYED CROW IS BRAN
Well first of all if it's not Brynden there's really only one other person it could be. But let's dig in - first to the other weirwood, the sapling in Jon Snow's dream. He sees (as the chart says) a weirwood that is "no more than a sapling" that is growing as Jon watches, and directly has Bran's face...and three eyes. And while the ravens certainly love to say weird shit, I think it's very notable that they don't start saying his name until ADWD - imo, it's Bran getting stronger, figuring out easier how to communicate.
Later on, Theon has a very similar experience where he sees Bran's face in the weirwood tree in Winterfell, and hears the maester's ravens saying his name and calling for him to go to the weirwood. Both of these in conjuction make me believe that
the weirwood with the twisted mouth is different than the sapling weirwood -> they are not the same person, and potentially not aware of each other
the sapling weirwood is the three eyed crow
the ghost attempting to talk to Theon is not Brynden - this is obvious, because it's Bran, that's directly stated, but Bran is using not just his face in the weirwood but the ravens saying Theon's name to communicate...
And if you look a the chart, this happens with Jon as well!
Bran is trying to reach them both and he's doing it through dreams, the Winterfell weirwood specifically, and the maester's ravens
And following the "suspicious behavior" train of thought, Coldhands himself is incredibly suspicious. "Your monster, Brandon Stark." is so weird and the ravens echoing really brings that point home. Coldhands is directly talking about the three eyed crow....and calls him "bran's monster." But no one else does this. Brynden doesn't. Leif doesn't. It's because the three eyed crow is a monster Bran made, and Bran made himself into a monster.
The one dream that involves the three eyed crow that I haven't talked about yet is Jojen's - the dream of the crow trying to free the winged wolf. I think there's two ways to see this and the first is obvious: it's Bran trying to free himself. I've gone into here and a few other places why I think Bran is time traveling, and I think it's likely this is future, Three Eyed Crown Bran in all his monstrous glory, attempting to free our current, chained to the ground Bran. But even discounting the time travel aspect, I think it also makes a certain sort of sense if it's Bran trying to get himself free - it's right as Bran's third eye first starts creaking open that Jojen gets this dream! The chains are GREY and the Maesters are referred to as “grey rats” and the like due to their robes. And at this time, Bran is no longer dreaming because Maester Luwin is giving him droughts to cause dreamless sleep!
The other idea is that it's Bran attempting to free Jon Snow. We know Bran is attempting to communicate with Jon, as I outlined above, around this time. We know Jon has been ignoring all his wolf dreams, but especially during ACOK. And what is a Targaryen-Stark child but a wolf that can fly? The chains that the crow is trying to break are even Stark grey - Jon is chained down by his belief that he is Ned's Stark bastard, and not Lyanna and Rhaegar's!
I really think it could be either! But the three eyed crow, the crow trying to free the winged wolf, the crow that knows exactly how distracting Jaime is to Bran, the crow who interacts with Bran and teaches him to fly and shows him the North and the Others and Jon's death, rather than just calling his name? The crow that is described by Coldhands as "a friend" and "a monster" both? That's all Bran babeeee.
