You think Boeing lets any schmuck work on their aircraft? The airlines have Boeing certified and trained mechanics that work closely with Boeing to conduct maintenance and repairs. Apple, similarly, allows you to become an authorized maintenance service provider for their products.

In any case, the airlines are often not the owners either. A lot of new aircraft are actually leased from Boeing or Airbus. That's kind of how I view my cell phone really. A leased appliance that I'm happy to let the maintenance of which be someone else's problem.

Sometimes they do (cf Dreamliner battery problems).