But it doesn't matter the words your father spoke to you, the reality is he permanently and irrevocably damaged and lessened your mother's life. Forever.
His affair child is a permanent reminder of his appalling betrayal and the all the suffering your mother endured because of his choice to betray her. These boards and the entire internet is littered with the stories of how desperately unhappy people are when their husband or wife chooses to betray them.
Yes, I didn't suggest in any post your mother should make the choice for you and stated it was YOUR decision repeatedly.
But do not expect your mother to reframe her trauma to suit your half sibling or you at any point in her life. She probably can't and it's perpetuating the abuse your father inflicted on her by choosing to betray her to expect her to do that. It doesn't matter if it was a hundred years ago, your mum deserves your compassion for what your father chose to do to her.
You shouldn't expect or ask her at any time to want to see or talk to the reminder of his choice to betray her, she has every right in the world to say your half sibling has nothing to do with her - because she doesn't. And every right not to want to talk to her, see her and basically just try to forget she exists, if that is her way of coping with the memory of your father's choice to betray her.
Of course it's not your half sibling's fault, she lost out big time because of your father's choices. But she was NOT abandoned by you because you have absolutely no responsibility to her. None at all. Nor does your mother or your sister.
Your father was the only one who had a responsbility to her. It is all, every last bit of this sad tale, his fault. Not your mother's, sisters or yours. You are all women dealing with the selfishness, cruelty and betrayal of a man and the fall out of that.
Yes, I am sorry the letter went to your mum. What a horrible shock for your mum to have to deal with that letter, and on top of your father's death.
So yep, as per previous posts, I would just go and see her if you choose to, in the full knowledge it may or may not work out, and then briefly tell your mother and sister what you did without letting them attack you for it, as your boundaries and choices are your own. And theirs are their own.