In the history of Church dogmatics and Christian theology, the Holy Spirit is the oft-neglected Person of the Trinity. The Christological controversy resulted in seven ecumenical councils. The Holy Spirit is only briefly mentioned, but with only implied references to His Godhead (c.f. 1st Council of Constantinople). This is single example reflects a history of neglect.
This matters not only because the confessional logic never advances beyond sheer faithfulness, but because the Holy Spirit loses His personhood and becomes collapsed into Human structures. In the former case, sheer faithfulness is not a bad thing. But it's a little disconcerting. Many Baptists I've met continued the practice of baptism only because of Scripture command. There was no deep reason besides an artificial connection between water and faith. You get wet if you're serious about your commitment. But unlike St. Paul, who spoke of the need of baptism and its typology, most Baptists lay the emphasis on praying the prayer. I'm picking on Baptists, but they're not the only ones.
But sheer faithfulness is helpful to a point. When one wants to dare to engage in a defense of the faith, then baptism might just become one more weird thing and cast aside for more important matters. That's a shame, especially since Paul made effort to talk of the unity of baptism (i.e. "there is one baptism").
However, what is worst is the collapse of the Holy Spirit into Human structures. In Rome, this can be how the Magisterium functions. In confessional bodies, the Confession takes the place of the work of the Holy Spirit as the maintainer of unity. A friend of mind woefully repents of his false trinity of "Father, Son, and Self-Reliance", turning theological and pastoral work into one of sheer Human effort. Neither is the Holy Spirit a kind of Zeitgeist, a confirmation of what we want to be so (as the psuedo-charismatics of the apostate Episcopal church insist). The Holy Spirit is not a World Soul, a Church soul, or a philosophic or intellectual soul. He is God, truly and fully.
When the Apostles, among other Elders, met in Jerusalem, St. James, brother of the Lord, said that the decision seemed "good to us and to the Holy Spirit". There was a distinction that maintained a boundary that much of the West has forgotten. When the boundary is closed to hearing God's Word spoken anew, then protectiveness sets in. The Imperial Papacy is merely an extreme example of what might occur. The sort of Confessional brutality, leading to schism after schism among Protestants, was equally disturbed by a lack of the Holy Spirit.
Besides changing a creedal/worship document arbitrarily, the addition of the 'filioque' is a part of the problem. When a real dyad is built out of the Father-Son, where their differentiation on the processing of the Spirit is adumbrated, the Holy Spirit loses His significance and becomes similar to an energy force exuding out of God. I am not blaming the 'filioque' as the source of problems, it is rather an example of symptomatic general disregard.
Sadly, it's a neglect of the Holy Spirit, and His full Personhood and full Godhood, damages many parts of the Christian life. It can lead to moralism or a pessimistic antinomian approach. It can lead to an Ecclesial conquest (vis. an Imperial Church become Babylon) or an Ecclesial capitulation (i.e. the Church as an organ of the State or a sphere of the Society). It leads to antiquarian yearns for a past that never was or a constructed, anxiety-prone, future.
When the Holy Spirit is recognized as the Leader of the Church, the Wind that blows where He wills, and reveals Christ in the strangest ways, it results in good. There is an openness to hear the Word of God in other places and times. There is an openness to guidance that is not fatalistic. There is Human regeneration. There is organic, and real, ecclesial unity.
I am not arguing for Pentecostalism or Charismaticism. Those movements have some unhealthy impulses within them. But at least they have a certain disposition that is open to God continuing His work. Though they are not always willing to "test the spirits", their attitude is right.
If the Holy Spirit is just a tertium quid and you can not begin to address the question "Why Trinity? Why not Four, Five, Six etc.?". How does the Holy Spirit not become a mere 'etc.' in the Triune Life? Whether it is the politics of the Church (by this I mean the Church existing in the world), the forms in a particular church-community, how to read the Scripture, or the ethics practiced by Christians, the Holy Spirit guides us in all of these. He seeks only to bring Christ. If we have not the Holy Spirit, we have neither Christ nor the Father.
Holy Spirit, Lord of Life, fill our hearts, life us into Christ, and bring us to the Father of all Light.