How do advocates of an intermediate state explain that the Old Testament seems to be unaware of the existence of the Bosom of Abraham?

This is going to be an answer in outline, since I don't have time to research the supporting materials (which I know exist):

1. A lot development in beliefs about the intermediate state as well as the spiritual realm happened during the 400 years after Malachi (written around 445 BC). This becomes the contextual background of NT writers including Jesus himself.

2. Assertion from NT weighs more, part of the scripture interprets scripture principle and the progressive revelation principle. So Christians give greater weight to what Jesus, Paul, and John said. Hence we are not troubled by the lack of mention in pre-450 BC OT.

3. But OT+NT combined was never very explicit about it in the first place, giving us only hints. sola scriptura believers say it's enough and God (and Jesus) didn't want to reveal more. Roman Catholics and Eastern Orthodox believers place more currency on the early church tradition that continued the trajectory of the pre-Jesus development, resulting in greater refinement of the intermediate state.

How was an intermediate state, like the bosom of Abraham, not understood at the time, but was later on?

But as touching the resurrection of the dead, have ye not read that which was spoken unto you by God, saying, I am the God of Abraham, and the God of Isaac, and the God of Jacob? God is not the God of the dead, but of the living. And when the multitude heard this, they were astonished at his doctrine.

Something in hindsight can be so clear once someone explains it.

To answer the OP, the doctrine is there and clear, but not revealed until Christ.

This is similar to other doctrines. For example, here is another.

By faith Abraham, when he was tried, offered up Isaac: and he that had received the promises offered up his only begotten son, Of whom it was said, That in Isaac shall thy seed be called: Accounting that God was able to raise him up, even from the dead; from whence also he received him in a figure. Heb 11:17-19

Or this that speaks spiritually to the same:

And the clean person shall sprinkle upon the unclean on the third day, and on the seventh day: and on the seventh day he shall purify himself, and wash his clothes, and bathe himself in water, and shall be clean at even. Num 19:19

Much was not fully revealed in Old Testament times - and how could it be ? For Christ was not yet come.

Justification by faith was hinted at in Abraham but not fully expressed until after Christ suffered and died and the gospel was fully conveyed : the righteousness of God himself unto all and upon all that believe, through faith of Jesus Christ.

Reconciliation was not fully revealed, a new headship under Jesus Christ in a new humanity.

Remission of sins was not known in its full experience, sins were not utterly eradicated until Christ came and suffered and bore them, himself, in his body on the tree.

Nor, until Jesus Christ himself told us of the rich man and Lazarus in hades could we ever know of such a place, such a great gulf, such torments of the wicked and such a place of bliss as the bosom of Abraham.

No wonder is it that such things were as yet unrevealed - until Christ came.

But some had remarkable insight, granted to them, individually :

For I know that my redeemer liveth, and that he shall stand at the latter day upon the earth: And though after my skin worms destroy this body, yet in my flesh shall I see God: [Job 19: 25-26 KJV]

Job saw even beyond the state of hades, beyond the corruptions of the grave, beyond a resurrection and saw into another state altogether : he and his Redeemer, upon an earth, risen past death.

Therefore my heart is glad, and my glory rejoiceth: my flesh also shall rest in hope. For thou wilt not leave my soul in hell; neither wilt thou suffer thine Holy One to see corruption. [Psalm 16:9 KJV]

And David sees, in the Spirit, that his own flesh shall rest in hope and he will not be left in hades. For God's Holy One shall not see coruption.

The hope is not a division of hades, with a gulf to separate, the protection of Abraham's bosom and the torment of disembodiment and flame. The hope that is hoped is for new heavens and a new earth and for glory yet to be revealed : in final resurrection and a new creation.

We may not fully understand (nor can we) the disembodied state of death, but we are given sure and certain promises regarding the resurrection and eternal glory in the world to come.

That is the hope of those who follow Jesus Christ out of this world and into another yet to come.