Why do commercial flights avoid overflying Shandong Province?
This is a guess, but it may be to avoid the restricted area ZSR-522 that covers airspace just north and east of the Shandong Peninsula. It's hard to say exactly, but it looks like your flight path more or less skirted around it, and the IFR chart suggests that the flight might have followed A326 east and then south from the ANRAT waypoint.

The Chinese AIP says (or said; that link may be out of date) that ZSR-522 is only up to 18000ft so in theory it's still well below typical airliner cruising levels. I've read several times, though, that the Chinese military control their airspace very tightly and they typically don't allow direct flight paths so perhaps they simply insist that aircraft must follow the airways because of 'national security' or some similar reason. Example:


  And the military's control of nearly all the airspace between Chinese
  destinations means that flights within China, even by the favored
  national carriers, sometimes must fly indirect routes that are the
  equivalent of going all around the city on a ring road.