Yeah, Takumi does some hate-inducing stuff in Conquest, and there's the prologue chapters when he gives you that rude introduction, but the thing to keep in mind is that he suffers from an inferiority complex. As the middle child, he feels like no one cares about him, so when Corrin returns to Hoshido, he comes out the way he does because he felt neglected for so long with the Hoshido family worrying all about Corrin for so many years. Yes, he's a jerk to Azura initially, but he just doesn't like the idea how she is a replacement for Corrin and could continue the tradition of the extra family individual that gets all the attention.
To be honest, his initial reaction is probably the most human reaction out of all the Hoshido siblings. I mean, sure, you'd probably be all warm and happy if you saw a long-lost sibling again, but at the same time, what if the sibling was raised all this time by your enemy nation? You'd probably feel hesitant approaching the individual because of where they are coming from.
Going into Conquest's plot, he does attack you several times, but put yourself in his shoes. How would you react when your mother gets killed and your sibling abandons your family after a tragic event like that? You would probably become overwhelmingly furious.
And going back to the inferiority complex thing, it's exactly that reason that he is very easy to be possessed or manipulated by some unknown force: Iago in Birthright and Anankos in Conquest. Heck, Azura even points out after Chapter 13 that it isn't the Takumi she remembers; she knows Takumi was always a kind dude and not someone who would raise his voice or threaten anyone. Corrin asks if his decision to join Nohr is part of the reason why Takumi is becoming insane, and Azura says it could be. It does make sense when your link about it. Feelings from an inferiority complex, trauma over his mother's death, upset from Corrin's decision is making him lose his sanity, hence why he think killing Corrin might help him gain some composure. Of course, a loss of sanity is also the perfect opportunity for a demon to possess him.
Meanwhile in Birthright, he does a fill a little reassured that you join Hoshido's side, but he still suffers the middle child inferiority complex and hence why he gets possessed in Chapter 10. Azura saves him, and she, Hinoka, and Sakura are able to reassure him that they care about him. Even Corrin reaches out to him and this is Takumi's response.
He opens up and feels bad for what he said. That shows he has a kind side. Play through the game, and you'll see he's a kind dude in many of his supports, and even acts like a relatable dork in different parts of the story. There's also the ending to Conquest, where he gives a heartfelt confession for all that he has done.
Cut to the late game in Birthright when Iago manages to manipulate Takumi in Chapter 25, and once again thanks to Azura singing, gaining his senses, and hearing the reassuring words of everyone as his mind is freed, he delivers this line.
Quite the development if you ask me. You can even sense his growth with his supports with Hinoka.
Then there's his B-Support with Kiragi, where Takumi says one of the deepest and most relatable lines out of any character in the game.
I don't know about you, but I'm sure we all had feelings like that at some point in our lives.
All in all, Takumi is probably one of the deepest characters in all of Fates, and it's because of that he's my favorite character in all of Fates. All it takes is to give him a second chance to really understand him.