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