Yet few people would call an airplane with auto-pilot a "self-flying plane", even with advanced autopilot that can take off and land itself.

Autopilot is understood to mean a limited computer assist that can help with routine tasks, but not expected to allow a plane to fly without a pilot.

What more would a plane need to be "self-flying" beyond taking off, navigating, and landing?

The ability to handle unusual emergency or unexpected conditions - the same thing that's needed to call a car "self-driving".

It means staying on a set course. Not the same thing, right?

A "fancy autopilot" can take off and land.