As a believer in what some choose to term 'the tripartite nature of humanity' I would say that Jesus omits nothing in the text by only making reference to two nouns:
And be not afraid of those killing the body, and are not able to kill the soul, but fear rather Him who is able both soul and body to destroy in gehenna. [Matthew 10:28 YLT]
The soul is the vehicle of the living spirit. Both are immaterial. But the soul has function and capability. The living spirit is associated with that immaterial soul.
To cover all the necessary texts would require pages of input. My book 'The Son of Man' contains the necessarily lengthy study and is available free of charge as a download, see my profile for the link to my website.
Hebrews 4:12 expresses this:
... the dividing asunder of soul and spirit, and of the joints and marrow, ...
wherein the soul is likened to joints (the Greek word used for 'joint' is associated with a chariot) and the spirit is likened to the living tissue of marrow within the bone (from which comes red blood).
Thus the mechanical aspect expresses structure and function (the soul) and the living aspect conveys life itself (the spirit).
Paul expresses the same when he speaks of "spirit soul and body", 1 Thessalonians 5:23. One could punctuate that as spirit: soul and body, for the immaterial spirit resides in the immaterial soul and the organic life (spirit) resides in the body of flesh.
The expression 'body soul and spirit' confuses the whole issue and is not the way Paul chooses to word the matter.
Thus Jesus conveys, in Matthew 10:28, a dichotomy of that which is visible and tangible in the world (the body) and that which is immaterial and spiritual (the soul) within which the immaterial living spirit is present, in life or in death. (I repeat, in life and in death.)
The study of this demands considerable attention to, and understanding of, the Greek word psuche which is a broad concept and is translated into English as both 'life' and 'soul' depending on context, indicating how broad that spectrum is.
It is also necessary to fully appreciate the reference in Genesis to God making Adam a 'living soul' and breathing into him the 'breath of life'; and necessary to appreciate the words nephesh (not exactly 'soul') and ruach (breath, wind and spirit - similar to the Greek pneuma) in Hebrew.
Some suppose (wrongly) that God (somehow and unrighteously, for Adam possessed no inherent righteousness) breathed the 'Holy Spirit' into Adam.
This misconception clouds much of the discussion about this matter.