The short answer is that, from the Catholic Church’s perspective, God did not merely choose to reveal Himself as Father. Rather, He is Father by nature, in two respects: through the eternal generation of the Son in the Holy Trinity, and in His act of creating, especially the creation of those creatures that are “in his image” (see Gen. 1:26). Therefore, human fatherhood is best viewed as a partaking in the Divine Fatherhood of God, one that, by God’s design, is most apt for understanding Divine Fatherhood.
The use of the term “Father,” therefore, is not to be viewed as a concession to the patriarchal culture of the time, but as revealing the very nature of God.
Divine Fatherhood in the Scriptures.
As the question says, the Scriptures—both the Old and the New Testament (the New more explicitly than the Old)—take great pains to reveal God as Father.
In the Old Testament, God is referred to numerous times as “Father,” although the name of Father never clearly acquires a meaning beyond the metaphorical. For example, there is Deuteronomy 32:6:
Do you thus repay the LORD, you foolish and senseless people? Is not he your father, who created you, who made you and established you? (ESV).
or Isaiah 63:16
For you are our Father, though Abraham does not know us, and Israel does not acknowledge us; you, O LORD, are our Father, our Redeemer from of old is your name.
(For quite a good overview of God the Father in the Old Testament, see “The Fatherhood of God” on http://biblestudytools.com.)
In the New Testament, of course, it is well known that Jesus calls God his Father and instructs us to do the same. (See, above all, the Lord’s Prayer,
Matthew 6:9-13 and Luke 11:2-4; also John 20:17.)
Notably missing—in either Testament—is any reference to God as “mother,” despite the fact that God clearly has attributes that might be considered “maternal." For example, God comforts His people as a mother comforts her child (Isaiah 66:13); like a nursing mother, He will never forget His people (Isaiah 49:15); He longed to gather Jerusalem as a hen gathers her chicks (Luke 13:34). (See my source for these examples.) There are also multiple references to God’s “tender compassion” (e.g., Luke 1:68-79).
Is the reference to the Father necessary, or merely a cultural concession?
Having established the fact that God revealed Himself as “Father,” the question now is establishing why.
The reason comes from the very nature of the Holy Trinity. As the Catechism of the Catholic Church (CCC) puts it,
Jesus revealed that God is Father in an unheard-of sense: he is Father not only in being Creator; he is eternally Father in relation to his only Son, who is eternally Son only in relation to his Father: “No one knows the Son except the Father, and no one knows the Father except the Son and any one to whom the Son chooses to reveal him” (No. 240, quoting Mt. 11:27).
God is Triune—that is, Father, Son, and Holy Spirit—by nature from all eternity. The Fatherhood of God is not simply a “persona” that God assumed; it is He very Being. (He is also Son and Holy Spirit, of course.) He is also “Father” in the sense that His act of creating resembles the generation that is proper to human fatherhood (strictly speaking, as I mention below, it is more a question human fatherhood resembling God’s Fatherhood).
As the Catechism explains, human fatherhood is an image of or participation in God’s Fatherhood:
By calling God “Father”, the language of faith indicates two main things: that God is the first origin of everything and transcendent authority; and that he is at the same time goodness and loving care for all his children (No. 239).
It is clear, therefore, that God is Father by nature, and calling Him so does not depend on merely cultural considerations.
Order of being, order of knowledge
At this point, it is useful to distinguish two aspects of this problem: there is an ontological order (the order of being), which deals with things as they really are, independently of our knowledge; and there is the order of knowledge, which deals with how we come to know that reality.
It should be noted that God is perfectly ineffable; that is, we are able to know Him only to the degree that He reveals Himself to us (either through His effects in creation, or through Revelation). As the Fourth Lateran Council (1215) puts it,
We firmly believe and confess without reservation that there is only one true God, eternal infinite (immensus) and unchangeable, incomprehensible, almighty and ineffable, the Father and the Son and the Holy Spirit; three persons indeed, but one essence, substance or nature entirely simple (Canon 1; translation from CCC 202).
Jesus revealed that within the Godhead there is a unique relationship between himself and the Father. Trinitarian theology teaches that the relations from Father to Son and Son to Father (and, of course, from the Holy Spirit to Father and Son) are identical with the Persons themselves:
252 The Church uses (I) the term “substance” (rendered also at times by “essence” or “nature”) to designate the divine being in its unity, (II) the term “person” or “hypostasis” to designate the Father, Son and Holy Spirit in the real distinction among them, and (III) the term "relation" to designate the fact that their distinction lies in the relationship of each to the others (No. 252; also St. Thomas Aquinas describes the Persons as “relations subsisting in the Divine Nature”; see Summa theologiae Ia, q. 30, a. 1 co.)
The relation proper to the Father is called generation (see CCC No. 254) or Fatherhood. This type of fatherhood is, of course, the very first in the order of being, and all creaturely fatherhood partakes in this one.
However in the order of knowledge, the question becomes, “What means did God have at His disposal in order to reveal His Divine Fatherhood to us”? God, as I have mentioned, is ineffable; hence even when He reveals Himself to us, He must make use of those aspects of reality that resemble Him. Indeed, things resemble Him precisely because they are effects of His creative power. (As Aquinas puts it, omne agens agit sibi simile, every agent produces an effect similar to itself; for instance, fire makes other things hot. See, e.g., S.Th. Ia, q. 44, a. 2 arg. 3.)
In other words, when Jesus called God “Father” (and based on that, the Church Fathers called the relation of the Father “generation”), he was employing human fatherhood as an analogy, so that we could have a glimpse into the ineffable mystery of the Holy Trinity.
The name “Father” signifies the essence; the maternal characteristics are attributes.
I mentioned that God does possess many attributes can be expressed by maternal images. As the Catechism says,
God’s parental tenderness can also be expressed by the image of motherhood, which emphasizes God’s immanence, the intimacy between Creator and creature. The language of faith thus draws on the human experience of parents, who are in a way the first representatives of God for man. But this experience also tells us that human parents are fallible and can disfigure the face of fatherhood and motherhood (No. 239).
Nevertheless, as should be evident from God’s ineffability, God greatly transcends human reality:
We ought therefore to recall that God transcends the human distinction between the sexes. He is neither man nor woman: he is God. He also transcends human fatherhood and motherhood, although he is their origin and standard: no one is father as God is Father (No. 239).
However, when Jesus revealed God as Father, he did so in a way that signifies the very essence of God (namely, the Father as eternally begetting the Son, and God as the creator of all things). “Father,” therefore, is appropriate as a name for God, especially since man knows about the Fatherhood of God only by analogy with human fatherhood.
On the other hand, the “maternal” attributes of God do not signify the essence. (Yes, in the ontological order, God’s attributes are identical with Himself; but we, human beings with limited intellects, are compelled to distinguish between the two in our minds—that is, in the order of knowledge.)
This distinction explains why Jesus commanded us to call God “Father,” as I mentioned above, but not “mother”. This must not be understood in any way as a denigration of human motherhood: rather, human fatherhood and motherhood, from our point of view, signify different aspects of our understanding of God.
The related question: why did God make human fatherhood resemble His own?
The question thus is better reformulated in this way: why did God create human fatherhood to resemble His own Fatherhood? Any attempt to answer this question will necessarily enter into the realm of speculation, but presumably, it is in order that human beings, through human fatherhood, would be able to learn something about Divine Fatherhood—since that is the only way we could know about it.
However, God’s Fatherhood, in the order of being, is eternal and independent of His creatures.