This is a good question, but I am afraid I cannot answer the question with certainty. God does not tell us why he reveals certain things to us at certain times and conceals others. I can speculate about this, and I will do a bit of that below, but please be aware this is simply my thoughts and please only take if for what it is worth.
First of all, Jesus did give something pretty close to a “full discourse” on the Holy Spirit in both John 14 and John 16. Also, he talked about the Father all the time. I am not sure what you expect–that Jesus give a philosophical/theological lecture, perhaps? Jesus did not give theological discourses. He taught largely through parables. But Jesus clearly and unambiguously declared himself to be God (John 8:52-50 and John 10:29-33, for example). You will see that in the John 10 passage, Jesus talks about God the Father, as he did often. As for the Holy Spirit, like I said, Jesus spoke at length about him, using the personal pronoun, and equating him with the Father and himself in John 14 and 16. Therefore, I am going to partially deny your second premise that Jesus did not give a “full discourse” about the triune nature of God, but leave you to think about that.
On the other hand, your first premise, that there is only a rather limited discussion of what we call trinity in the Old Testament–that the Prophets do not give “complete details” about the triune nature of God, is correct. There is nothing in the Old Testament that clearly lays out the existence of three “persons” in the godhead. We can see hints of the Son in several places (Psalm 110:1 Psalm 2:7 and many more), and there is actually a fair amount of mention of the Holy Spirit in the Old Testament (Ezekiel 37:4-10, Ezeiel 36:24-28 and many more), but it is not in a form that caused the Jews to really “get” it. Why is this? Like I already said, I do not know. Perhaps God chose to wait to make it fully clear because, before Jesus came, to talk about a three-in-one God would have been confusing at best and may even have pushed the Jews toward a kind of polytheism. Like Jesus said in the Sermon on the Mount, in Matt 5:17, he is the fulfillment of the Law and the Prophets. He is the one who completed the message and made it clear. Why did God choose to only tell us this in somewhat veiled forms before Jesus? I already gave you my speculation that it may have tempted the Jews to overly speculate and even to move toward polytheism, but God may have had other reasons I do not know.