That would only be true if the Father isn’t also God in trinitarian theology; which, duh.
But no, the other phrase (God the Son) is never used in scripture so far as I know, whereas Son of God is–though rarely. There seems to have been a reason why it was rarely used, too: it’s the sort of title that would have been, especially along with Jesus’ other claims, easily understood to mean… what? Example John 10:33, where the title “Son of God” isn’t even being used, but the same implications are being deployed.
Now, it might be argued that, in this example, Jesus’ enemies are creatively misconstruing him by putting together his claims, including that God is his Father, in a false fashion. (And there are quite a few claims being made here in this scene, not even counting other scenes in GosJohn, not even counting other scenes in the Synoptics, not to even mention Acts, the Epistles and RevJohn afterward…) To which charge, by the way, Jesus doesn’t answer by correcting them as to the ludicrousness of their charge, but rather by quoting a Psalm (82) where Elohim Who possesses all nations and judges the earth arises to take His stand in the congregation of Elohim where He judges in the midst of unjust elohim who do no know or understand and walk about in darkness; of which unjust elohim, the Psalmist (and/or possibly Elohim Himself) declares “I on my part said you are elohim, and all of you are sons of the Most High, nevertheless you shall die like men and fall like any one of the princes.”
When Jesus quotes the first part of verse 6 to them, he (or He!) is treating them with a standard rabbinic rebuke form, where they are expected to pick up the implications and especially to finish out the statement: he expects them to recognize that he is claiming they are putting themselves in the place of the unjust elohim against whom the one and only Most High Elohim is taking His stand against them in the midst of His own congregation–while still distinguishing himself personally compared to the Father.
So in a rabbinically witty roundabout way, Jesus is not answering “Of course I am not making myself out to be God Most High, I am only a creature like you nothing more.” He is answering, “Yep, I’m doing exactly that, you got it right, and you of all people should have been expecting this to happen but instead you’re rejecting Me just like the Psalm prophecied the lesser sons of the Most High would judge unfairly against the Most High Who judges them when He comes to take His stand among them.” He’s saying this in a way that still keeps the distinction of persons, Father and Son, but he’s still saying he’s really He the Most High.
It’s things like this, plus all the other claims Jesus makes about Himself (and which are made about Him by NT authors), for which Christians later call Him not only Son of God but also God the Son, even though “God the Son” doesn’t show up as a phrase in the NT.
But leaving aside that particular bit of exegesis, it’s still true that in the Gospel narratives, claiming to be the Son of God was a much riskier thing than to be claiming to be the Son of Man. At the very least, the narrative evidence is that claiming to be himself “Son of God” would be too easily misconstrued as meaning that Jesus was making himself out to be God. It’s one thing for some (over?)-eager disciples to call him “Son of God” (Matt 14:33, 16:16; John 1:49), or for demons to be trying to declare this (which tends to get them quickly muzzled by Jesus, Matt 8:29; Mark 3:11, 5:7; Luke 4:41, 8:28), or for the king of demons to tempt him to prove it (Matt 4:3, 6; Luke 4:3, 9), or for a pagan soldier to say something along this line in admiration (Matt 27:54; Mark 15:39). But when Jesus himself claims this about himself, or agrees that he is, he (or He) tends to get into major trouble with the religious experts (Matt 26:63; 27:40, 43; Mark 14:61; Luke 22:70; John 19:7).
And I don’t think it takes a whole lot of really close reading to figure out why the religious experts think this amounts to reprehensible blasphemy–they’re probably worried that the rabble who know not the Torah and are accursed, will start treating Jesus as being the Memra of God (from the rabbis’ own Aramaic targums at the time) Who was not only with God but actually was God, by Whom all things have come into existence and apart from Whom nothing has come into existence, in Whom exists and came into being the Life that is the Light of men, through Whom grace and truth (or joy and reality) came, the only-begotten God in the heart of the Father Who unfolds the Father that we may see the unseeable God and gaze upon His glory (the Glory as of an only-begotten from the Father full of joy and reality).
Which, considering that Jesus had also answered his rabbinic accusers (seeking his death because in calling God his own father he was thus making himself equal with God) that just as the Father has life in Himself, even so He gave to the Son also to have life in Himself and thus to give life to whomever He wishes, so that the dead in the tombs who hear the voice of the Son of God shall live, just as the Son has authority to execute judgment because he is the Son of Man (John 5:18-29, definitely reffing Daniel there), it must have seemed to the rabbis and chief priests like a very real risk Jesus’ followers would come to think of him that way. (Though whether we ever did or not, who can say? )
Anyway, my point is that even though the phrase “God the Son”, per se, is never (so far as I know) used in scripture, the scriptures themselves testify that for Jesus to call himself the Son of God could (at the very least) easily be construed (especially in combination with other things he, or He, was saying) as being tantamount to claiming to be God the Son: equal to God, in a Shema unity with God the Father, even though personally distinct in comparison to the Father.