How do Latter-day Saints explain testimonies of profound experiences with the Holy Spirit in other denominations?

We generally don’t feel a need to explain them.

The Church of Jesus Christ of Latter-day Saints doesn’t claim a monopoly on experiences with God (see Alma 29:8). The church does, however, make the significant claim of being the church directly organized & led by Jesus Christ and invested with His authorized priesthood power. That’s a bold claim. I understand why adherents of other faiths do not like it. If the claim is an invention of man, it’s absurd. If it comes from God, it doesn’t matter if it sounds absurd. The means offered by Latter-day Saints for testing the claim is seeking a witness directly from God.

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Some doctrinal background

In discussing what is meant by the gift of the Holy Ghost, there may be value in offering some doctrinal background on ordinances and covenants. The Church of Jesus Christ of Latter-day Saints teaches that there are 5 ordinances of salvation (see here):

1. Baptism
2. Confirmation (including conferral of the gift of the Holy Ghost)
3. Ordination to the Melchizedek Priesthood (for men)
4. Temple Endowment
5. Temple Sealing

Each of these ordinances is accompanied by one or more covenants (for sake of simplicity let’s define covenant here as a two-way promise with God), and we believe that (for people who are morally accountable) they are required for exaltation. Each of these ordinances must be performed by one holding the requisite priesthood office, under the authorization of one holding priesthood keys.

For those interested in the implications my comments carry for faith/works, I have presented a more detailed discussion of grace and covenants here.

The point of the covenants is not to complete a checklist that thereby earns our salvation; the point of the covenants is the transformation they produce in us. God (who is a really good parent) asks us to do things because doing those things will help us. With each covenant or set of covenants, we promise things to God and He promises things we need—things we could never earn on our own—that enable us to progress and develop.

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The gift of the Holy Ghost in broader context

Other posts have discussed differences between the gift and power of the Holy Ghost and links have been shared for further reading. I’d like to focus in on just one big-picture principle here.

The fruit of a tree (from the plant’s perspective) is not an end in itself—it’s a means for growing a tree. So too, I propose, is the fruit of the Spirit. It may impress, but that’s not its fundamental purpose. I am in fact wary of those who use gifts of the Spirit to draw attention to themselves (instead of to God).

The ordinances & covenants noted above are not ends in and of themselves, but means to an end. The God who designed tiny seeds that can transform into majestic trees wants to carry out an even greater transformation in His children. The power that comes into our lives (and beyond) through these ordinances (see Doctrine & Covenants 84:19-20) is essential for our exaltation. And so, I respectfully conclude that while the gifts of the Spirit and the power of the Holy Ghost are indeed manifested to many—regularly—the full transformative power of this and the other saving ordinances is only available through covenants instituted by God and performed by His power.

Several of the promises associated with baptism and confirmation were described by the prophet Alma:

9 Yea, and are willing to mourn with those that mourn; yea, and comfort those that stand in need of comfort, and to stand as witnesses of God at all times and in all things, and in all places that ye may be in, even until death, that ye may be redeemed of God, and be numbered with those of the first resurrection, that ye may have eternal life—

10 Now I say unto you, if this be the desire of your hearts, what have you against being baptized in the name of the Lord, as a witness before him that ye have entered into a covenant with him, that ye will serve him and keep his commandments, that he may pour out his Spirit more abundantly upon you? (Mosiah 18:9-10)

The promise at the end of verse 10 conveys nothing about measuring time. The Spirit will be poured out more abundantly through these covenants. How much of the time do different people feel guided by the Holy Ghost? I propose that this question is unanswerable except by God. I am not God ^{[citation needed]}, so I will not venture a guess. What the Book of Mormon does teach, though, is that pouring out the Spirit in greater abundance (than otherwise available) is a gift from God that is dispensed via His covenants, entered into by His power.

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Conclusion--personal note

The prediction in the OP--as stated--is not a prediction I as a Latter-day Saint would make.

It is one thing to judge how well an organization’s doctrine aligns with sacred texts; it is quite another to judge the authenticity of another person’s relationship with God. As a flawed, fallen, imperfect human being, full of motes, beams, and plenty in between, it is not my place to do the latter.