You’ll notice that the Sign of the Cross is “in the Name of the Father, and of the Son, and of the Holy Spirit.” Three persons, but the Name, singular, not the Names, plural. That is because the Name of the Trinity is “Father, Son, and Holy Spirit.” The nature of Undivided Unity that is the essential aspect of the Trinity means that we have to use a singular pronoun for the Trinity, too. The only way you would use “they” is if you were to refer to two Persons and mean to discuss them separately from the third Person, so that you were not referring to the Trinity, but talking specifically about the Persons as distinct so as to express some issue of the relationship. That is, if you meant to talk about the Son and the Holy Spirit but not the Father, you would use “they”.
I tell my religious education students that English is not well-equipped to express the Trinity in pronouns. For instance, all sentient beings, all persons, that we know of are “he” or “she”, not “it”. Among creatures normally expressed in English, there are no sentient persons without a gender. Also among creatures, there is no unity of persons such as is present in the Trinity. The closest thing is an organization, and that, more and more, lands you back into the land of the non-person “it”. Even in the world of science fiction, a sentient “it” is very “other” than we are. It puts a distance in.
Therefore, we have some choices. Which is more vital to expressing God: no gender or sentient with personhood? Sentient and personhood wins, so we can get rid of “it” as a possible pronoun for the Trinity. Which to use after that: “they”, “he”, “she” or a random choice? As noted above, the pronoun must be singular, so “they” is out. Well, it was not that long ago that the name for a species as a whole also named the male: horse, fox, man, and so on. So although the Trinity has no gender, the best choice of pronoun is “He”, since to use “She” would imply a female gender or a feminine identity. This is the traditional choice. (Traditionally the Church, which has no gender but was considered to have a feminine identity in relationship with God, is referred to as “she”, but that usage is fading.) As for the “random” choice, just try it. It is never random.
On the topic of the Holy Spirit, in Hebrew there is the Shekhinah, the indwelling presence of God, which is grammatically a feminine noun. Although people do it, and I can see the argument, the problem with using the grammatical feminine when alluding to this noun in order to speak specifically of the Holy Spirit is that English has pretty much lost all feminine and masculine nouns, save those which refer to actual genders. Referring to the Holy Spirit as feminine might work grammatically in other languages that have feminine nouns that refer to things not having a female gender. It does not work that way in English. If you use the feminine pronoun, you will be taken to mean a female. Unfortunately, more and more, if you use the masculine, you are taken to mean a male. This is too bad, because it makes it extremely difficult to use pronouns to talk about God without getting sucked into a “battle of the sexes”. It also sticks us with awkward sentence constructions where one has to use “he or she” or else artificially change to the plural “they” in order to not sound as if one refers to one particular gender. On the other hand, one can hardly talk of feminine and masculine, as one once did between God and the Church, without being taken to mean literally male and female. I still do so, but I realize I do it at my peril. Oh, well.
But that is another soapbox. The best we can do for now is to use “He”, with the traditional understanding that a grammatically masculine noun is appropriate for a God with no gender, whether referring to one particular Person of the Trinity, or to the Trinity as a Unity of Persons. That is how the grammatical experts have explained it to me; I’m not a grammatical expert.
Let us hope that between our political obsession with physical gender, the loss of grammatical gender, and the prevailing thought that complementarity must automatically branded as sexist, the language doesn’t evolve so as to make the question utterly impossible to answer. I do know this: if someone comes up with the pronoun equivalent of “Ms.” to deal with this issue, I may choke.