Though it is harder to follow the arcs on such a long passage, v.5-18 comprise a complete unit that I think is best understood as a whole. Summary: v.5-8a, God did not subject the 'world to come' to angels but to man. Psalm 8 is quoted as an integral part of this assertion, which is then explained in the remainder of the chapter. In other words, v8b-18 is an exegesis of Psalm 8 in light of the incarnation, crucufuxion, and resurrection of Jesus. (Most commentators place v.8b-9 with v. 5-8a as one unit. I do not disagree that vv. 8b-9 offer an explanation of the author's use of the Psalm 8 quote. However, I see vv.10-18 as just as much a part of the explanation of Psalm 8 as are vv.8b-9.) v.8b-9, This subjection is total, nothing is excepted. However, we do not presently see everything in subjection. But what we do see is Jesus - in accord with Psalm 8, made lower than the angels in order that he might taste death for mankind. This understanding of the psalm is supported by the following bilateral arrangement: v.10-13, It is fitting that Christ joins the company of men, as brothers. And so (v.14-15), since the children are flesh, he took on human flesh in order to destroy death and deliver us. v.16, Surely it is not angels he helps, but men - the sons of Abraham. This stands as the foundation at the heart of the entire discussion. v.17-18, Therefore he had to be made like his brothers to be a faithful high priest, suffering temptation and able to offer genuine help!
Οὐ γὰρ ἀγγέλοις ὑπέταξεν τὴν οἰκουμένην τὴν μέλλουσαν, περὶ ἧς λαλοῦμεν.
For it was not to angels that God subjected the world to come, of which we are speaking.
διεμαρτύρατο δέ πού τις λέγων• τί ἐστιν ἄνθρωπος
It has been testified somewhere, “What is man,
ὅτι μιμνῄσκῃ αὐτοῦ,
that you are mindful of him,
ἢ υἱὸς ἀνθρώπου
or the son of man,
ὅτι ἐπισκέπτῃ αὐτόν;
that you care for him?
ἠλάττωσας αὐτὸν βραχύ τι παρʼ ἀγγέλους,
You made him for a little while lower than the angels;
δόξῃ καὶ τιμῇ ἐστεφάνωσας αὐτόν,
you have crowned him with glory and honor,
πάντα ὑπέταξας ὑποκάτω τῶν ποδῶν αὐτοῦ.
putting everything in subjection under his feet.”
ἐν τῷ γὰρ ὑποτάξαι [αὐτῷ] τὰ πάντα οὐδὲν ἀφῆκεν αὐτῷ ἀνυπότακτον.
Now in putting everything in subjection to him, he left nothing outside his control.
Νῦν δὲ οὔπω ὁρῶμεν αὐτῷ τὰ πάντα ὑποτεταγμένα•
At present, we do not yet see everything in subjection to him.
τὸν δὲ βραχύ τι παρʼ ἀγγέλους ἠλαττωμένον βλέπομεν Ἰησοῦν διὰ τὸ πάθημα τοῦ θανάτου δόξῃ καὶ τιμῇ ἐστεφανωμένον,
But we see him who for a little while was made lower than the angels, namely Jesus, crowned with glory and honor because of the suffering of death,
ὅπως χάριτι θεοῦ ὑπὲρ παντὸς γεύσηται θανάτου.
so that by the grace of God he might taste death for everyone.
Ἔπρεπεν γὰρ αὐτῷ, διʼ ὃν τὰ πάντα καὶ διʼ οὗ τὰ πάντα, πολλοὺς υἱοὺς εἰς δόξαν ἀγαγόντα τὸν ἀρχηγὸν τῆς σωτηρίας αὐτῶν διὰ παθημάτων τελειῶσαι.
For it was fitting that he, for whom and by whom all things exist, in bringing many sons to glory, should make the founder of their salvation perfect through suffering.
ὅ τε γὰρ ἁγιάζων καὶ οἱ ἁγιαζόμενοι ἐξ ἑνὸς πάντες•
For he who sanctifies and those who are sanctified all have one source.
διʼ ἣν αἰτίαν οὐκ ἐπαισχύνεται ἀδελφοὺς αὐτοὺς καλεῖν
That is why he is not ashamed to call them brothers,
λέγων• ἀπαγγελῶ τὸ ὄνομά σου τοῖς ἀδελφοῖς μου,
saying, “I will tell of your name to my brothers;
ἐν μέσῳ ἐκκλησίας ὑμνήσω σε,
in the midst of the congregation I will sing your praise.”
καὶ πάλιν• ἐγὼ ἔσομαι πεποιθὼς ἐπʼ αὐτῷ,
And again, “I will put my trust in him.”
καὶ πάλιν• ἰδοὺ ἐγὼ καὶ τὰ παιδία ἅ μοι ἔδωκεν ὁ θεός.
And again, “Behold, I and the children God has given me.”
Ἐπεὶ οὖν τὰ παιδία κεκοινώνηκεν αἵματος καὶ σαρκός,
Since therefore the children share in flesh and blood,
καὶ αὐτὸς παραπλησίως μετέσχεν τῶν αὐτῶν,
he himself likewise partook of the same things,
ἵνα διὰ τοῦ θανάτου καταργήσῃ τὸν τὸ κράτος ἔχοντα τοῦ θανάτου,
that through death he might destroy the one who has the power of death,
τοῦτʼ ἔστιν τὸν διάβολον,
that is, the devil,
καὶ ἀπαλλάξῃ τούτους, ὅσοι φόβῳ θανάτου διὰ παντὸς τοῦ ζῆν ἔνοχοι ἦσαν δουλείας.
and deliver all those who through fear of death were subject to lifelong slavery.
οὐ γὰρ δήπου ἀγγέλων ἐπιλαμβάνεται
For surely it is not angels that he helps,
ἀλλὰ σπέρματος Ἀβραὰμ ἐπιλαμβάνεται.
but he helps the offspring of Abraham.
ὅθεν ὤφειλεν κατὰ πάντα τοῖς ἀδελφοῖς ὁμοιωθῆναι,
Therefore he had to be made like his brothers in every respect,
ἵνα ἐλεήμων γένηται καὶ πιστὸς ἀρχιερεὺς τὰ πρὸς τὸν θεὸν
so that he might become a merciful and faithful high priest in the service of God,
εἰς τὸ ἱλάσκεσθαι τὰς ἁμαρτίας τοῦ λαοῦ.
to make propitiation for the sins of the people.
ἐν ᾧ γὰρ πέπονθεν αὐτὸς πειρασθείς,
For because he himself has suffered when tempted,
δύναται τοῖς πειραζομένοις βοηθῆσαι.
he is able to help those who are being tempted.