Is there a prayer that is said by others on behalf of someone who is about to undergo a dangerous (non-cosmetic) surgery? Is the מי שברך the most appropriate?
I would recommend Refuah Shlema. It comes from Numbers 12:13 when Moses asked God to heal Miriam, "Please God, heal her." From my understanding, our sages have taught us that this did not ask for complete healing, but for simple healing. Thus adding the adjective "shelma," asks for a prayer of healing to the greatest extent.
Mi-sheberakh avoteinu v'imoteinu, Avraham v'Sarah, Yitzhak v'Rivkah, Ya'akov, Rachel v'Leah hu y'varekh et (recite the English and Hebrew name) v'yavi aleihem refuat hanefesh u'refuat haguf yachad im kol cholei amo Yisrael. Barukh atah Adonai, rofeh ha'cholim.
May the One who was a source of blessing for our ancestors, bring blessings of healing upon (recite the English and Hebrew name), a healing of body and a healing of spirit. May those in whose care they are entrusted, be gifted with wisdom and skill, and those who surround them, be gifted with love and trust, openness and support in their care. And may they be healed along with all those who are in need. Blessed are You, Source of healing. Amen.