Just my personal viewpoint - based on what my own J.W. family's reaction to your wording would be.
An editing of your letter:
"I hope this correspondence finds you all well. I'm writing here to reach out for a relationship with my family. I love you all, I worry about you, and want nothing but the best for you as I am sure you feel the same for me. I believe we are all grown up enough to show respect for one another as human beings. Kids after all are only kids once in their life.
I was hoping my children could have relationships with Aunts, and Uncles that I was not privileged to have growing up. Much of this due to petty differences between our parents and their siblings. I now know where that judgmentalism stems from. I know you all have your opinions about what happened with my family. You are likely partially correct. Almost losing Adin to leukemia played a large part.
All of this being said I value peace as well, and hope we can all continue to be family. I have waited to write all of you until Adins treatment neared it's completion. As I am planning a party to celebrate the completion of his treatment that I would like you all to attend.
I respectfully request that you keep this within our immediate family so as not to further harm our Mom, Me, Ruthy, and My Children. The only grandparent my children can have a relationship with. I am not my child's owner just a steward."
Sometimes less is more! I hope you achieve your goal, and have a wonderful family celebration.