Not really a placebo, but broken collarbones are an example of this. The most common injuries can be healed with just a sling and rest. With surgery it makes sense with compound fractures or bones that are overlapping significantly or not where they need to be as it might not heal at all with a sling. Compared to just healing with a sling, surgery has increased risks from complications, the metal plate that can cause discomfort and a noticeable scar. With surgery the healing time is usually faster too