That is not necessarily true. Energy is a property of all fields. In QM (more specifically quantum field theory) the electron is seen as an excitation of an underlying field. This field does have a zero point energy so there is energy even at absolute zero. In fact absolute zero means absence of energy from thermal motion - that is the random motion of small-scale particles. It does not imply zero energy, just zero thermal energy. It would be difficult to produce a situation where a system at absolute zero had actually zero energy. It makes no real sense to speak of temperature for massless particles (this is not strictly true, but such particles will always have a definable energy that would still exist at zero temperature). For massive particles you would always have gravitational potential energy — all massive bodies attract each other. You also would have rest energy (the very name tells you that energy does not require motion). This is more commonly known (outside of relativity) as mass, a form of energy as given by Einstein’s most well-known equation E=mc^2.
You certainly know much more in this arena than I do. Thanks for the constructive information, FRiend.