There are verses that throw a pretty big wrench in this perspective though.
John 14:21 Jesus says that He will manifest HIMSELF unto those who love Him and keep His Commandments (which are the Father's 10 Commandments [Jn. 15:10; Matt. 23:1-3]). When asked to clarify what He means by this in verse 22, Jesus reiterates by explaining that when He says He will manifest "HIMSELF" to us, what He means is that He and the Father will come to make their home with us. Note: zero mention of a Holy Spirit because God IS a spirit and Rom. 8:9-11 makes the case that JESUS is the Holy Spirit.
This blows the idea of the Father and Son being separate right out of the water. If God were three persons, Jesus would have certainly mentioned the 3rd "person" somewhere, some way, in John 14:21-23.
There is no verse that "blows the idea of the Father and Son being separate right out of the water." Nor should we presume what Jesus would have said based on a preconceived notion we have. Context is everything and one verse taken on its own, disregarding everything else that is said about the Father, Son, and Holy Spirit, is being taken out of context. I have found that that is the main issue with anti-Trinitarians--verses are taken one a time, piecemeal, but that is mere proof-texting.
Consider what Jesus says right in John 14, verses 16-17:
Joh 14:16 And I will ask the Father, and he will give you
another Helper, to be with you forever,
Joh 14:17 even
the Spirit of truth, whom the world cannot receive, because it neither sees him nor knows him. You know him, for he dwells with you and will be in you. (ESV)
Repeated in verse 26:
Joh 14:26 But
the Helper, the Holy Spirit, whom
the Father will send
in my name, he will teach you all things and bring to your remembrance all that I have said to you. (ESV)
First, notice that the language of "another Helper" means one that is similar to but distinct from Jesus himself. Second, we see all three persons mentioned distinctly in verse 26. Indeed, throughout the whole NT, all three persons are continually and consistently mentioned as separately, which would make no sense and be pointless if they weren't all actually distinct from each other.
Now notice verse 28:
Joh 14:28 You heard me say to you, ‘I am going away, and I will come to you.’ If you loved me, you would have rejoiced, because
I am going to the Father, for the Father is greater than I. (ESV)
And, on it goes with the distinction of Father and Son being maintained, throughout the whole NT:
Joh 15:1 “I am the true vine, and my Father is the vinedresser.
…
Joh 15:8 By this
my Father is glorified, that you bear much fruit and so prove to be
my disciples.
Joh 15:9
As the Father has loved me, so have I loved you. Abide in my love.
Joh 15:10 If you keep my commandments, you will abide in my love, just as I have kept
my Father's commandments and abide in his love.
…
Joh 15:21 But all these things they will do to you on account of my name, because they do not know
him who sent me.
…
Joh 15:23
Whoever hates me hates my Father also.
Joh 15:24 If I had not done among them the works that no one else did, they would not be guilty of sin, but now they have seen and hated
both me and my Father.
…
Joh 15:26 “But when
the Helper comes,
whom I will send to you from the Father, the Spirit of truth, who proceeds from the Father, he will bear witness about me. (ESV)
Joh 16:7 Nevertheless, I tell you the truth: it is to your advantage that I go away, for
if I do not go away, the Helper will not come to you. But if I go, I will send him to you.
Joh 16:8 And when
he comes,
he will convict the world concerning sin and righteousness and judgment:
Joh 16:9 concerning sin, because
they do not believe in me;
Joh 16:10 concerning righteousness, because
I go to the Father, and you will see me no longer;
…
Joh 16:13 When
the Spirit of truth comes,
he will guide you into all the truth, for
he will not speak on his own authority, but whatever he hears he will speak, and he will declare to you the things that are to come.
Joh 16:14
He will glorify me, for
he will take what is mine and declare it to you.
Joh 16:15
All that the Father has is mine; therefore I said that he will take what is mine and declare it to you.
…
Joh 16:17 So some of his disciples said to one another, “What is this that he says to us, ‘A little while, and you will not see me, and again a little while, and you will see me’; and, ‘because
I am going to the Father’?”
…
Joh 16:23 In that day you will ask nothing of me. Truly, truly, I say to you, whatever you ask
of the Father in my name, he will give it to you.
…
Joh 16:25 “I have said these things to you in figures of speech. The hour is coming when I will no longer speak to you in figures of speech but will tell you plainly
about the Father.
Joh 16:26 In that day
you will ask in my name, and
I do not say to you that I will ask the Father on your behalf;
Joh 16:27 for
the Father himself loves you, because you have loved me and have believed
that I came from God.
Joh 16:28
I came from the Father and have come into the world, and now I am leaving the world and going to the Father.”
Joh 16:29 His disciples said, “Ah, now you are speaking plainly and not using figurative speech!
Joh 16:30 Now we know that you know all things and do not need anyone to question you; this is why we believe
that you came from God.”
Joh 16:31 Jesus answered them, “Do you now believe?
Joh 16:32 Behold, the hour is coming, indeed it has come, when you will be scattered, each to his own home, and will leave me alone.
Yet I am not alone, for the Father is with me. (ESV)
Joh 17:3
And this is eternal life, that they know you, the only true God, and Jesus Christ whom you have sent.
Joh 17:4 I glorified you on earth, having accomplished the work that
you gave me to do.
Joh 17:5 And now,
Father, glorify me in your own presence with the glory that I had with you before the world existed.
Joh 17:6 “I have manifested
your name to the people whom
you gave me out of the world. Yours they were, and
you gave them to me, and they have kept your word.
Joh 17:7 Now they know that everything that you have given me is from you.
Joh 17:8 For I have given them the words that you gave me, and they have received them and have come to know in truth that
I came from you; and they have believed
that you sent me. (ESV)
There is simply no way to understand what Jesus says in these passages other than that the Father is not the Son and neither are the Holy Spirit. Jesus repeatedly states that he "came from God" and is going "to the Father." Notice in John 15:24 Jesus says, "both me and my Father," and in 16:32 that "I am not alone, for the Father is with me." The plain meaning of those verses is that they cannot be the same person.
John 17:5 is interesting considering what Yahweh says in Isaiah 42:8:
Isa 42:8 I am the LORD; that is my name;
my glory I give to no other, nor my praise to carved idols. (ESV)
We
must take all verses together as a complete whole and
then make sense of what they say, for the full revelation of who God is.