John's gospel is open and transparent as to Who "God" is. The words "God" and "Father" are essentially interchangeable. In the KJV translation it is true that "Father" appears in 103 verses (122 times) in John as compared to "God" appearing in 69 verses (83 times)--so the word "Father" is of more frequent occurrence (based on a case-sensitive search).
Jesus prays to his "Father" in John 17, tells us that "the Father" is the one we should worship (John 4:21-23), and says plainly that the Father is both his God and our God (John 20:17).
But if John 3:16 were considered at all ambiguous as the identity of the One sending Jesus, we have only to consult the many other references in John plainly establishing this.
Sent of the Father
That all men should honour the Son, even as they honour the Father. He that honoureth not the Son honoureth not the Father which hath sent him. (John 5:23, KJV)
I can of mine own self do nothing: as I hear, I judge: and my judgment is just; because I seek not mine own will, but the will of the Father which hath sent me. (John 5:30, KJV)
But I have greater witness than that of John: for the works which the Father hath given me to finish, the same works that I do, bear witness of me, that the Father hath sent me. (John 5:36, KJV)
And the Father himself, which hath sent me, hath borne witness of me. Ye have neither heard his voice at any time, nor seen his shape. (John 5:37, KJV)
As the living Father hath sent me, and I live by the Father: so he that eateth me, even he shall live by me. (John 6:57, KJV)
And yet if I judge, my judgment is true: for I am not alone, but I and the Father that sent me. (John 8:16, KJV)
I am one that bear witness of myself, and the Father that sent me beareth witness of me. (John 8:18, KJV)
For I have not spoken of myself; but the Father which sent me, he gave me a commandment, what I should say, and what I should speak. (John 12:49, KJV)
He that loveth me not keepeth not my sayings: and the word which ye hear is not mine, but the Father's which sent me. (John 14:24, KJV)
Then said Jesus to them again, Peace be unto you: as my Father hath sent me, even so send I you. (John 20:21, KJV)
Sent of God
For God sent not his Son into the world to condemn the world; but that the world through him might be saved. (John 3:17, KJV)
For he whom God hath sent speaketh the words of God: for God giveth not the Spirit by measure unto him. (John 3:24, KJV)
Jesus answered and said unto them, This is the work of God, that ye believe on him whom he hath sent. (John 6:29, KJV)
Jesus said unto them, If God were your Father, ye would love me: for I proceeded forth and came from God; neither came I of myself, but he sent me. (John 8:42, KJV)
And this is life eternal, that they might know thee [the Father] the only true God, and Jesus Christ, whom thou hast sent. (John 17:3, KJV)
There are twice as many references to the "Father" sending Jesus, and yet, the Father and God are one and the same. In Jesus' prayer, he plainly calls the Father "the only true God." This would be a false statement if Jesus were also the true God, because Jesus is not the Father.
It is entirely improper for anyone to suggest a better wording to an inspired writer. To do so is to adopt the attitude of "higher criticism" in which one supposes he or she knows better than the Bible writer what he should have said. In the case of John 3:16, John is quoting Jesus--so this may not have been John's choice of wording at all, but that of Jesus.
If, as Jesus taught, the Father is God--the only true God, then there is nothing extraordinary about God having sent His son, as recorded in John 3:16-17.
But John's gospel teaches plainly that it is not just "God" who loves us, but "the Father himself."
The Father's Love
Now before the feast of the passover, when Jesus knew that his hour was come that he should depart out of this world unto the Father, having loved his own which were in the world, he loved them unto the end. (John 13:1, KJV)
He that hath my commandments, and keepeth them, he it is that loveth me: and he that loveth me shall be loved of my Father, and I will love him, and will manifest myself to him. (John 14:21, KJV)
Jesus answered and said unto him, If a man love me, he will keep my words: and my Father will love him, and we will come unto him, and make our abode with him. (John 14:23, KJV)
For the Father himself loveth you, because ye have loved me, and have believed that I came out from God. (John 16:27, KJV)
Now, looking at that last verse, there does appear to be a requirement to receive the Father's love. It is that we love Jesus, and believe that he "came out from God." If Jesus were "God," then how could he have "come out from" God?
That is the question of some importance here. If we want to receive the Father's love, it is important that we believe the truths that Jesus taught. If we love Jesus, we will believe and obey his commandments--and the Father himself will love us.
Side note: Many do not seem to realize that the word "Godhead" is singular. It is an archaic form akin to "maidenhead," which in most English words today would have shifted from the "-head" suffix to the "-hood" suffix. Therefore, references to "the Godhead" are, ironically, referring to a singular entity, the plural of which would be "Godheads."