Is a complex airplane endorsement required to act as PIC of a plane without all three elements of the "complex airplane" definition?
No, you do not need a complex airplane endorsement for this airplane because it doesn't have retractable gear, unless it is a seaplane.

All three elements are required for that endorsement to be necessary for landplanes.

A jet-engine is not a controllable-pitch propeller, therefore it would not require a complex endorsement even if it has retractable gear and flaps. 

However, the above would require a type-rating because it is a jet.

An airplane with a controllable-pitch propeller and retractable landing gear that isn't equipped with flaps would also not fit the definition of complex, so you are absolutely right.

See §61.31(e) Additional training required for operating complex airplanes

The definition for complex is in the Airplane Flying Handbook Ch 11, 


  “A complex airplane is defined as an airplane equipped with a
  retractable landing gear, wing flaps, and a controllable-pitch
  propeller. For a seaplane to be considered complex, it is required to
  have wing flaps and a controllable-pitch propeller.”


The definition for complex as shown in 14 CFR §61.1(b)(iii):


  "Complex airplane means an airplane that has a retractable landing
  gear, flaps, and a controllable pitch propeller, including airplanes
  equipped with an engine control system consisting of a digital
  computer and associated accessories for controlling the engine and
  propeller, such as a full authority digital engine control; or, in the
  case of a seaplane, flaps and a controllable pitch propeller,
  including seaplanes equipped with an engine control system consisting
  of a digital computer and associated accessories for controlling the
  engine and propeller, such as a full authority digital engine
  control."