Braime is never going to happen. Jaime says "I love Cersei" in 6x07 and gets her PREGNANT circa 7x04 => married as hell. He loves/admires Brienne in his way but they'll never be together.

He gives her gifts and then sends her away

A gift that represents his heart. Both Nik and Mark Mylod have stated this as a fact which makes me believe it says so in the script notes. In the books, Jaime giving Brienne her Oathkeeper is paralleled with the tale of the Maiden given the Perfect Knight a blade named the Just Maid after he had stolen her heart. The sword was a token of her love.

He's a man of means and arguably a great deal more free than most in Westeros, but he never drops everything and sets off to find his Brienne.

He sets off at the end of season 7. In the books, he literally abandons his army to take off with Brienne.

7x07: "I'm loyal to Cersei."

Ten minutes later he abandons Cersei. The script notes stating he is never looking back again.

After being relieved of his position as Kingsguard and Tommen's death, he was arguably free to abandon Cersei. He didn't. He stood at the head of their armies. He thought she was cray-cray, but he never left.

He abandons Cersei much earlier in the books. Given the Stoneheart arc was cut. Jaime was written to stay with his sister much longer. But ultimately he does leave.

Yes, he's headed North to keep his oath and try to prove himself to Brienne, but their love doesn't have the lifespan intensity of the "one heart, one flesh, one soul" union of Jaime x Cersei.

Wrong again. Jaime's relationship with Cersei was all about her. She doesn't respect him or show him any compassion. With Brienne he will experience true love.

Saving Brienne (twice) and loving her was the best thing he did in his life, but it doesn't make up for his crimes against the gods: treason, incest, murder, adultery, etc. You can't help who you love but you can help what you do about it.

Loving Brienne has nothing to do with making up for his crimes. It's about loving Brienne. However, as GRRM has said he is exploring redemption with Jaime and he wants to believe that redemption is possible.

/r/freefolk Thread Parent