Why does the Book of Revelation allocate such a subordinate role to the Holy Spirit?

The Book of Revelation is not an overtly trinitarian work in the sense that it never mentions the three persons of Trinity as such. (The reference in 1:4-5 is not to the Holy Spirit but the "seven spirits who are before his throne.") That is not to say that its author would deny the existence of the Trinity. But the Trinity as such only does not play an overt role in this particular scripture.

Why is the Trinity never overtly mentioned? The answer might be phrased as another question: "Why does it need to be?" The Holy Spirit is not mentioned in several other NT scriptures such as Galatians, Colossians, I Timothy and Philemon. Neither is the Holy Spirit worshiped as such in these and other NT writings. For that matter several scriptures do not speak of Jesus as the object of worship. So the absence of the term and the Spirit's apparently subordinate role should not be particularly problematic for believers in the Trinity. This is especially so if one accepts that John of Patmos is the same writer as John the Apostle. After all, the HS is not mentioned in several Pauline epistles, but it is definitely spoken of in others.

However, there is reason to doubt that John of Patmos and John the Apostle are the some person. The other works by "John" have a remarkable unity of language and theology, especially focused the importance of love, rebirth and the sacraments. But these concepts are not emphasized in the Book of Revelation. It is full of apocalyptic visions and dire warnings. Forgiveness is hard to find. The saints even pray for vengeance against their persecutors: '“O Sovereign Lord, holy and true, how long before thou wilt judge and avenge our blood on those who dwell upon the earth?” (6:10) This is a stark contrast to the attitude of John's Gospel:

he breathed on them, and said to them, “Receive the Holy Spirit. If you forgive the sins of any, they are forgiven.” (John 20:22-23)

Indeed, the word "forgive" cannot be found in the Book of Revelation. Neither can the word "mercy." The absence of the Holy Spirit as the third person of the Trinity in Revelation can be explained by similar absences in other NT scriptures. But the seeming absence of the fruits of the Spirit is harder to explain.

Subordinate can mean "under the authority of" or it can mean "of lesser value or importance". In this answer I hope to demonstrate that the latter of these two is reasonably impossible and that the former is properly Trinitarian.

If the Word of God goes forth and accomplishes all that it was sent to accomplish, then it is easy to see the Word as subordinate in the former sense. If that Word represents not only the expression but also the actual rational mind of God then it is easy to understand that such subordination does not preclude ontological sameness: There can be no discrepancy between who God is and what God thinks and says (He is not divided).

The Spirit of God is that by which He accomplishes all of His will; the action of God. The power that He exerts in His action is then subordinate in the same sense as His Word and possesses like ontological sameness as the Word: There can be no discrepancy between who God is and what God does (He is not divided).

Each of these are solid starting points to develop a trinitarian theology wherein an infinite, eternal God shares infinite, eternal, ontological sameness with His Word and His Spirit both of whom go forth in subordination and accomplish His will. The Trinity proper is not in defense in this question, and it is well treated elsewhere, but the ontological sameness of God and God's action (Spirit) provides a good basis to answer the question at hand.

Why does the Book of Revelation allocate such a subordinate role to the Holy Spirit?

Nowhere in Scripture is it the Holy Spirit's role or intention to be the center of attention. The Holy Spirit testifies always of the Lord Jesus Christ, both in the Church and in the world:

But when the Comforter is come, whom I will send unto you from the Father, even the Spirit of truth, which proceedeth from the Father, he shall testify of me: - John 15:26

Nevertheless I tell you the truth; It is expedient for you that I go away: for if I go not away, the Comforter will not come unto you; but if I depart, I will send him unto you. And when he is come, he will reprove the world of sin, and of righteousness, and of judgment: Of sin, because they believe not on me; Of righteousness, because I go to my Father, and ye see me no more; Of judgment, because the prince of this world is judged. - John 16:7-11

For this reason it is not surprising that the Holy Spirit is not specifically mentioned at each and every instance of action or influence throughout Scripture. Understanding that, at the very most basic level, all of the sovereign plans of God throughout creative and redemptive history are wrought through His Spirit should allow us to perceive and understand the Spirit's presence regardless of specific mention:

And the angel that talked with me came again, and waked me, as a man that is wakened out of his sleep, And said unto me, What seest thou? And I said, I have looked, and behold a candlestick all of gold, with a bowl upon the top of it, and his seven lamps thereon, and seven pipes to the seven lamps, which are upon the top thereof: And two olive trees by it, one upon the right side of the bowl, and the other upon the left side thereof. So I answered and spake to the angel that talked with me, saying, What are these, my lord? Then the angel that talked with me answered and said unto me, Knowest thou not what these be? And I said, No, my lord. Then he answered and spake unto me, saying, This is the word of the LORD unto Zerubbabel, saying, Not by might, nor by power, but by my spirit, saith the LORD of hosts. - Zechariah 4:1-6

This is Zechariah's vision of the "true" Church called out from every tribe, tongue, and people and it comes to be by the Spirit of the Lord of Hosts. This Church was in view in the very act of creation as Paul indicates in Ephesians when he reveals the union of Adam and Eve as a type of Christ and the Church. Revelation, being an apocryphal account of the consummation of Zechariah's vision, is no less brought to pass by the same Spirit.

One may ask after a Scriptural demonstration that the Spirit is a "third person" of the Trinity (there is no shortage of words applied here) but a lack of overt mention of the Spirit in any particular book does not constitute a subordinate (read as less important) role since everything that the Father and Son accomplish is done by and in the Spirit.

It is not the desire of the Spirit to garner worship toward any other than the Lord Jesus Christ and, through Him, the Father. The apparent absence of overt mention of the Spirit in the book of Revelation, especially in scenes depicting worship of God and of the Lamb, is actually good proof that the Spirit is present and accomplishing the will of God.

The presence of the Holy Spirit in Revelation is pervasive. Three times it mentions "the seven spirits of God", such as here:

Then I saw a Lamb, looking as if it had been slain, standing at the center of the throne, encircled by the four living creatures and the elders. The Lamb had seven horns and seven eyes, which are the seven spirits of God sent out into all the earth. (Revelation 5:6)

Collectively, these seven spirits describe the sevenfold nature of the Holy Spirit. They are a delegation of the Holy Spirit sent by Jesus, who is the one who said he would send the comforter. The seven spirits each manifest the wisdom and power of the Holy Spirit in a different way, and are described in Isaiah:

The Spirit of the Lord will rest on him—
    the Spirit of wisdom 
    and of understanding,
    the Spirit of counsel 
    and of might,
    the Spirit of the knowledge 
    and fear of the Lord (Isaiah 11:2)

Numerologically, each member of the Trinity is most closely associated with a distinct number:

1. The Son is two (2). Jesus has two natures, human and divine. As the Word, he is divided into Old and New Testaments. As to his visits, he comes twice, first as suffering servant, second as conquering Messiah.

2. The Father is three (3). In Revelation, the Father is "him who is, and who was, and who is to come". He represents eternity, all time, past, present and future. The Father also is supreme over the Trinity.

3. The Holy Spirit is seven (7). In addition to the seven spirits, the number seven dominates the structure of Revelation, with seven churches, seals, trumpets, bowls, thunders, and more.

So when you see the number seven in Revelation, the Holy Spirit is at work.