This is such a fabulous chant, such a delight to sing, that I wouldn't fuss over it, just do it! Goodness, given the chants to do something exquisite, don't grouse. It marks Trinity as a special day, well worth doing.

This is not just in the Gregorian Missal - it's in the Graduale Romanum (remember that the GM is basically the GR for Sundays and Solemnities), it has the authority of Church Law behind it, until it is officially suppressed. You are not here replacing the Proper with a "mere hymn"; you are given the choice to do the Gradual or this Hymn (both are the same text, one longer than the other), the choice between two different propers. The GIRM refers to replacing the responsorial psalm/Gradual with a vernacular song/hymn from a hymnal.

News flash: hymns are not the enemy.

I confess I find this very remarkable and, seemingly, inexplicable.

That's precisely what I like about this quirky option! Illogical. Serendipitous. Like grace. Deo gratias.

Perhaps because it’s here!:

http://www.e-codices.unifr.ch/en/sbe/0121/373/small

Einsiedeln, Stiftsbibliothek, Codex 121:

“This Codex comprises the oldest complete surviving neumed mass antiphonary; it includes assorted appendices (such as Alleluia verses, Antiphons and Psalm verses for the Communion Antiphons). Because the mass antiphonary is complete, the manuscript remains important to this day as a resource for Gregorian chant research. The second part of the codex contains the Libyer Ymnorum, the Sequences of Notker of St. Gall. Recent research has established that the codex was written in Einsiedeln itself (in about 960-970), most likely for the third abbot of the cloister, Gregor the Englishman.”

Its location in the Einsiedeln manuscript c. 965 A.D. does not tell us exactly where it occurred in the 10th century liturgy. It seems to be among various hymns, sequences and antiphons (it's hard to read so please correct me if I am wrong). At some point in history it ended up in the Mass of Saturday in Ember Week of Advent, after the fifth (!) reading (see the Liber Usualis).

To the heart of your question: when this Mass was dropped in the revised calendar, we may suppose that whoever redistributed the chants did not want this gem to be lost, and so gave it a new home as an alternate setting to a nearly-identical text. (In the Graduale Triplex there is an indication that the ancient form of the Graduale for Trinity was of similar length.) Who made the call? I wonder if someone else on the forum knows. It is noted in the Ordo Cantus Missae, which was the Vatican’s instruction for the redistribution of chants which governed the revised Graduale.

So who do we thank for this blessing? An anonymous monk of Solesmes, perhaps? But it was clearly part of the project of redistributing the chants.

The Canticle of the Three Young Men (plus a Trinitarian doxology) is an Old Testament canticle which points to the Trinity. The textual consideration and prior liturgical occurrence would seem to justify its seemingly odd placement in the reordered Graduale. Not only does it have a liturgical pedigree at Holy Mass from the above citations, but it is regular liturgical fare in the OF Liturgy of the Hours also (Sunday Morning Prayer, where it is called a canticle).

That said, it's alien enough to typical parish liturgical praxis (if there is such a thing) that, unless the priest is planning to bring attention to it in his homily it's difficult to consider preparing it--at least for the proposed liturgical placement in the OF liturgy. At the very least there should be a well-established use of the Gradual before attempting to use this hymn/canticle as an option.