In which case God must not be intrinsically Father, but only accidentally so (in the philosophical sense of ‘accident’).
That may not necessarily be a problem: God’s husbandship, even if God is trinitarian, is accidental that way, even though the husbandship is predicated on what God essentially is. But if we’re down to only talking about modal operations of God, we still have left over talking about what God essentially is. (Though there are some theologians, even inconsistently so among trinitarians , who say that we cannot possibly talk about what God essentially is in Himself. But without such a root then we have no ground for talking about God modally, either.)
The language apart from other contexts could only mean that, yes. (Or perhaps in context with other contexts!–not to salt the pizza too much on my side. )
Mainly the trinitarian case (rigorously speaking) appeals to the sonship mentioned in scriptures for purposes of arguing exegetically for a distinction of the persons. Which has an obvious relation to the Virgin Birth as well: Jesus could be born of a virgin by the miracle of God and so be Son of God in that specially unique sense as well as in the sense you’re talking about without sharing a full Shema unity with the Father (thus being fully divine).
This does however lead to observing that the ESV and the NIV are not, in the places you mention, translating the Greek quite right. They’re leaving out the ‘genes’ of the ‘monogenes’: that’s an awfully strong term to be using in regard to a one and only son of God, when something like “one and only” would have sufficed. (But there are other contexts around that statement in GosJohn, too, which indicate John means something more than only one-and-only. )
The only problem with this (which trinitarians certainly don’t deny) is that the characteristics of God ascribed personally to the Son (in the NT, and to the Angel of the Presence/Face/YHWH in the OT) are rather more than any merely created creature ought to have. And those characteristics are being ascribed to that person (and by that person sometimes), not merely to (and by) the person of the Father.
This has a bearing on the question of worship, too, which is ascribed personally to the Son as well as to the Father. The situation is not cut quite in the way you later suggest:
I will hazard some guesses as to why NT (and in their own way OT) authors focus more on worship of the Son than the Spirit (though not entirely neglecting the other).
1.) It wasn’t religiously problematic for a Jew to say they were worshiping the Spirit of God and to thus mean God fully God (not some mediator who wasn’t God but who somehow directed worship to God.) This would have made immediately sufficient enough sense to a Gentile, too, without much effort.
2.) On the contrary, while a religiously devout Jew might (or might not) accept that worship of a mere human king was tantamount to worshiping God, if the king was God’s Messiah (though by the 1st century this was certainly a huge problem for rabbis who typically tried to avoid it in order to avoid idolatry), worshiping a spirit who wasn’t God would have been obviously to worship an angel at best. Two spirits should not be worshiped but God alone.
3.) So what is the average Jew (or Gentile for that matter), or even the educated one, going to think if the Spirit is explicitly emphasized as being a distinct person? Two spirits!–which would lead back to polytheism for converted pagans, and would be rejected as polytheism by devout Jews.
4.) Yet it is blatantly obvious that Jesus Christ was (and still is) a distinct person. Moreover, a person who died a death traditionally (even scripturally) considered to be cursed by God.
5.) So the first (and later) preachers have a major tactical problem: the distinction of the personhood of Christ cannot be denied, yet (assuming for purposes of discussion that ortho-trin is true and that they had some sufficient knowledge of this now to even try to promote the idea religiously) they know they ought to be promoting the worship of a man who was and is also (somehow, in a way unique to him alone among all other men or angels) God Most High YHWH ADNY ELHM.
They have a tough row to hoe already (as we say in West Tennessee). So it would make sense for them to concentrate on the tougher issue that has to be directly faced and dealt with, and not so much on the issue of the Spirit which would naturally only bring in even more problems. (Though they don’t entirely ignore worship of the Spirit either.)
Something very similar certainly happened in the official theological disputes of the 4th century onward: the Christology was focused on first, and then later the Pneumatology. That hardly means trinitarians (or other Christians for that matter!) weren’t worshiping the Spirit before then; if they hadn’t been, there wouldn’t even have been a dispute.