Here is a brief piece brought to my attention in a prior comment that might represent a common evangelical view; it criticizes use of "Mother" to refer to God the Father.
Some of her points are astute enough; who would disagree with these?
Because God is not literally a father (i.e., a man who procreates), God is, therefore, a father in a metaphorical sense.
It should also be noted that God’s fatherhood is not about gender. The divine nature is not sexual or gendered in any sense. Although the human nature of Jesus is gendered, the divine nature is not. The fatherhood of God is not tantamount to the inherent masculinity of God.
Because the name “Father” is metaphorical and not literal, it does not speak literally of God’s having a male or masculine nature.
We have a broad base of agreement here, where I would say so far she is exactly right. But then she goes on to say:
The New Testament view is unmistakable: God is the Father of our Lord Jesus Christ, and Mary was his (merely human) mother. And God is not only Jesus’ Father, God is “Our Father.” We have been adopted to “sonship” and are heirs of God, coheirs with Christ. This is the picture and terminology that the Bible uses to present the family relationship of believers to God and Christ. There is no place in this picture for a Mother God alongside or instead of a Father God.
I've put what seems to me to be her main point in a bold font. I presume what she means to say in the last sentence goes further than what she says elsewhere:
The picture of God as a mother is also present in Scripture, such as when Jesus describes himself as a mother hen. However, to make this observation is not to imply that the “father” metaphor is on a par with the “hen” metaphor.
After all, "not being on a par" might be taken to mean "being different". Surely the NT uses fatherhood metaphors for adoption with much greater emphasis than any motherhood metaphors--there is an unmistakable difference in emphasis.
But she wants to draw attention to more than difference in emphasis; maybe she means the fatherhood metaphor has a certain aptness that the motherhood metaphor does not? This even though she readily admits
...it is abundantly clear, especially in the Old Testament, that God is both mother and father to his people. This is rightly understood in a metaphorical sense, pure and simple. God is to us like a mother and like a father.
It comes down to this: "Father" is both a metaphorical descriptor and a metaphorical name, while "Mother" is merely a metaphorical descriptor. Hence:
However, God as the Father of Jesus Christ—as the first person of the triune Godhead—is not “Father” merely in the sense of a simple metaphorical descriptor. Here “Father” serves as a metaphorical name. (A metaphorical name is to be distinguished from a simple metaphor, a figure of speech used to describe one or more attributes of someone or something)....Although the Bible speaks of God in metaphorical imagery that is motherly and feminine, “Mother” is never used in Scripture as a name for God.
Well, that does not do the trick for me. It's interesting, but the problem I have is simply that it is too easy to turn a metaphorical description into a metaphorical name. "Ball and chain" or "dragon" for instance can do duty as a metaphorical decriptor, and then be used to refer as a metaphorical name. There is no special logical barrier to using descriptions as proper names.
Maybe the emphasis of her argument however falls on the phrase "used in Scripture." That is, we should only use those names for God used in Scripture. In that case, the real work in her case is supposed to be done via what seems to me to be a very dubious principle: if it is not done in Scripture it should not be done. In which case I ought not to address God as "God" or "Father"--these names being translations of what appears in Scripture properly speaking. Or I should not drive a car, type, or brush my teeth.
That brings me back to the question: how is "Father" apt, and "Mother" unapt, in such a way that "Mother" should not be used?