Should my family attorney run our succession conversations?

No. Even excellent estate attorneys are usually the wrong person to facilitate family succession conversations. The skills don't overlap. Estate work is precise, legal, and black-and-white. Family transition work is emotional, relational, and ambiguous. Kris Kluver, in The Dysfunctional Family Office, has Robert Mitchell name this directly. He likes his attorney. He needs his attorney for the legal mechanics. He also recognizes that the touchy-feely stuff isn't his attorney's wheelhouse. There's a second issue. The family attorney has usually worked with the founder for years and is perceived by the kids as the founder's guy, even when the attorney tries to be neutral. The kids won't trust him to facilitate a balanced conversation. Use your attorney for what they're built for. Bring in someone different for the family work.