Why did Jesus only say "ἐγὼ εἰμί" rather than, "ἐγὼ εἰμί ὁ ὤν"?
Why are you expecting Jesus to have said something He did not say? Why He did not add "ὁ ὤν" is best answered "He did not want to say it." Questions of "motive" (why) are often very hard to answer firmly and purely from the text.
Longer Answer Based in Exegesis
So the core statement is this (my translation and notes):
| a reference to... πρὶν = before | relative time is being made here Ἀβραὰμ = Abraham | a person long dead is being named here γενέσθαι = came to be (Mid. infinitive) | that long dead one's birth is made here ἐγὼ = I | Himself is made here εἰμί. = am (present tense) | His state of existence is summarized here
Jesus was making a relative temporal reference to His state of existence at the time of Abraham's coming into existence, as a support point about how He could know that "Abraham rejoiced" about Him (v.56), which knowledge the Pharisees questioned (v.57).
Note that Jesus does not use the past tense being verb εἰμί to describe Abraham (i.e. before Abraham existed), but the verb γίνομαι, which emphasizes the idea of coming into existence (i.e. before Abraham came to be). For Himself, he uses the present tense of εἰμί to refer to His state of existence at this time "before" Abraham's birth.
So Jesus is not focusing per se on either the name of God proper (such as Ex 3:14), nor what that name implies—eternality. Rather, His focus is on a localized span of time, namely before (and implied during from v.56) Abraham's lifetime, Christ already was existing. That is how He knows that Abraham rejoiced, because Christ was there at that time. That is the focus of His statement.
Of course, implied in His statement is that (1) He has existed at least before Abraham, and (2) He is still existing now. This does not of itself mean eternality, but certainly long life. But coupled with the Jewish understanding of Ex 3:14, and Jesus previously mentioning God as His Father (part of the whole previous context within ch. 8, especially v.42), they were well aware that at least He was making a statement of equality with God, if not also perceived as a direct reference to being God, and so sought to kill Him for what they believed was blasphemy (v.59).
Good question. It is because in John 8:58 Christ wasn't referring to the name of God but rather, to his own pre-existence as a " living person" not as a " plan" as Socinian Unitarians think.
Notice the context:
John 8:42 The Son proceeded and came (ἐξελθεῖν)from the Father.
John 8:58 Before Abraham came into existence, Jesus is already existing.
The Greek" ἐξελθεῖν" occurs in the Septuagint of Genesis 15:4 and it teaches a real birth from a parent.
Hence, in John 8:42, Jesus is telling us that he is God's Son by nature ( John 5:18) not by adoption and in John 6:46, it is clear that Jesus has seen the Father because he came from him, that is, that he is the Son. This reveals that Jesus is not just a mere man for no man has ever seen God in John 1:18 and yet Jesus, existing before Abraham came into existence, already saw God for he was in the beginning " the Word who was with God and the Word who was God"(John 1:1).
This is the prevailing Christology in the ancient church and undoubtly, the true Biblical doctrine of God for it does not entail any contradictions.