You should ask a number of surveyors what they include in their reports, and more importantly, what the exclusions are. Ask for a sample report in advance so you can see the level of detail they go into, and the typical caveats they include to protect themselves.
You may not need a structural engineer to check it out, unless there are question marks over some of the work. In many cases they will limit their scope to an opinion on the structural integrity of the building, but will not go into full snagging mode.