As a practical matter, the FAA might designate someone outside of Boeing to do the inspections (like maybe people from Lockheed or Grumman, who are not directly competing in the commercial airliner space).

When you take a checkride at the end of training to earn your pilot certificate, it's not an FAA employee riding with you... it's a "DPE" or Designated Pilot Examiner, who is a private individual designated by the FAA to conduct check rides (and who often charges you a fee to administer the test).

Occasionally an FAA employee will ride along with the student pilot and DPE to evaluate how the DPE is evaluating prospective pilots.