Although the precise vocabulary used may differ, the witness of the Holy Spirit is significant for Christians of many faiths.
Theory
We'll show this from 2 different angles:
1. Where is the foundation
We are to put our trust in the Lord over the wisdom of the world:
5 Trust in the Lord with all thine heart; and lean not unto thine own
understanding.
6 In all thy ways acknowledge him, and he shall direct thy paths.
(Proverbs 3:5-6)
Paul's approach to teaching was to rely on the convincing power of the Spirit:
And my speech and my preaching was not with enticing words of man’s wisdom, but in demonstration of the Spirit and of power (1 Cor. 2:4)
If the foundation of my belief in Christ is man's discoveries in archeology, history, textual criticism, etc. then I leave my theology at the mercy of the wisdom of men. Prevailing hypotheses in academia vary over time--should a devout Christian abandon belief in the New Testament every time Bart Ehrman is cited in peer-reviewed research, then switch back every time William Lane Craig is cited, then switch back...etc? If the predominant claims in academic journals should dictate what we believe about God, then everybody should have stopped believing in God when the Tubingen school imposed naturalism as the standard for New Testament research in the 1830s. Clearly most practicing Christians do not/have not taken this approach.
This is not to say that these other disciplines are without value, but they are insufficiently stable to provide a foundation (and Jesus had a few things to say about foundations--see Matt. 7:24-27).
What the world gives the world can take away. What God gives the world cannot take away.
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2. Trust in the New Testament apostles
The testimony of the apostles that Jesus died, was buried, and rose again (see 1 Cor. 15:3-4) is absolutely central to Christian belief. Why is it that we trust the apostles?
One of the common reasons given for trusting Peter & Paul is their willingness to die for their testimonies. This is indeed valuable evidence (since they--and especially Peter--were in a position to know firsthand if they had made it all up), but how do we know they died for their testimonies?
Our primary source is the historical record written by someone who was there at the time (Clement of Rome, see his Epistle to the Corinthians, chapter 5). But most Christians have never read Clement.
It appears that the most widespread and enduring reason given for trusting the apostles is that their words, preserved in the New Testament, have been ratified by the Holy Spirit in readers' hearts & minds.
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Theory Conclusion
The witness of the Holy Spirit is essential to Christians who do not wish to be "tossed to and fro, and carried about with every wind of doctrine, by the sleight of men, and cunning craftiness, whereby they lie in wait to deceive" (see Ephesians 4:14).
Application
Reasons were provided above for expecting that belief be grounded in direct revelation from the Holy Spirit. The degree to which what is expected in theory is in fact the case will vary based upon individual application.
Citations from several Christian denominations are provided below attesting to the essential nature of learning & accessing truth through the Holy Spirit.
From the Catechism of the Roman Catholic Church:
the Spirit will teach us everything, remind us of all that Christ said to us and bear witness to him. the Holy Spirit will lead us into all truth and will glorify Christ. He will prove the world wrong about sin, righteousness, and judgment (source, emphasis mine)
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From the Orthodox Church in America:
The Christian Church lives by the Holy Spirit. The Spirit alone is the guarantee of God’s Kingdom on earth. He is the sole guarantee that God’s life and truth and love are with men. Only by the Holy Spirit can man and the world fulfill that for which they were created by God. All of God’s actions toward man and the world—in creation, salvation and final glorification—are from the Father through the Son (Word) in the Holy Spirit; and all of man’s capabilities of response to God are in the same Spirit, through the same Son to the same Father...The Holy Spirit is the personal presence of the new and everlasting covenant between God and man, the seal and guarantee of the Kingdom of God, the power of the divine indwelling of God in man. (source, emphasis mine)
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An Anglican perspective (note this is an Anglican publication but not an official Anglican source):
the world needs to know that Jesus and the Father are one, which it can't without the witness of the Holy Spirit in the unity of the church. (source, emphasis mine)
And that also means that we who know we have been found by God need to view the world very cautiously, knowing that our standards of judgement are not good, knowing that we are looking for the unity of the Father and the Son, which we will only be able to see through the work of the Holy Spirit, whose job it is. (ibid, emphasis mine)
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Presbyterian:
By the work of the
Holy Spirit, the gift of faith is given so we see reality
in a whole new way. Our hearts, minds, and wills are
made new!
Theologians call this the witness of the Holy Spirit.
Presbyterians have always stressed that we come to
faith in Christ as Lord and Savior by God’s divine work,
through the Holy Spirit, and not by our own powers
of reason or intuition or experience...We recognize, receive, and believe in this gift by the work of the Holy
Spirit. The Spirit enables us to look at creation and perceive it as God’s work. (source, emphasis mine)
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The Church of Jesus Christ of Latter-day Saints:
We invite all men everywhere to read the Book of Mormon, to ponder in
their hearts the message it contains, and then to ask God, the Eternal
Father, in the name of Christ if the book is true. Those who pursue
this course and ask in faith will gain a testimony of its truth and
divinity by the power of the Holy Ghost. (See Moroni 10:3–5.)
Those who gain this divine witness from the Holy Spirit will also come
to know by the same power that Jesus Christ is the Savior of the
world, that Joseph Smith is His revelator and prophet in these last
days, and that The Church of Jesus Christ of Latter-day Saints is the
Lord’s kingdom once again established on the earth, preparatory to the
Second Coming of the Messiah. (source, emphasis mine)
I discuss my own personal testimony relative to this promise here.
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Pentecostal Tabernacle:
The primary purpose of the baptism of the Holy Spirit is to empower believers to witness for Christ (Acts 1:8). The Holy Spirit will honor and glorify Jesus Christ (John 15:26; John 16:14), and provides believers with guidance, comfort, and truth. (source, emphasis mine)
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Application Conclusion
Do all Christian faiths emphasize this matter equally or in the same way? No. For a point & counterpoint discussion regarding a competing perspective, see the section "The Bible demonstrates the viability of inspiration" in this post.
To the OP's point, although I am unable to identify any denomination that denies the importance of the witness of the Holy Spirit (I found some unofficial, individual perspectives that do), I am unaware of any denomination that focuses on testimony through the Holy Ghost to the same degree or in the same way as The Church of Jesus Christ of Latter-day Saints.