We first need a model for -o proagon- translated as "who goes beyond" (NRSV). It is hard to imagine what going beyond could mean. So I prefer the Lexicon translation "leads the way." and assume that John is referring to a leader or teacher in a church. That fits in with the warning "we who teach will be judged with greater strictness" (James 3:1). It also echoes the strong words against false teachers in several of the epistles.
Why should teachers be singled out in this way? We can picture a school in which students of all kinds are welcome, including the most ignorant and badly behaved. But for the teachers in that totally welcoming school qualifications and standards of behavior are required. One cannot have a teacher who wants to teach error, racist ideas, or behavior which is the opposite of what the school aims to inculcate.
With that model in mind, what kind of teacher does John want to exclude? -my menon en ty didachy tou Christou- one who does not remain in the teaching of the Messiah. The Great Commission requires enrolling by baptism all and sundry to begin learning in the schools of the Holy Spirit among all nations, and then "teaching them to obey everything that I have commanded you" (see models in Matthew 28:20). Whereas ordinary church members may be ignorant and badly behaved, teachers in the schools of the Messiah must know and impart all that Jesus taught in the Gospels about God the Father, God the Son, and God the Holy Spirit.
Unfortunately denominations often add to the basics of the Nicene Creed items which are by no means obvious from the Gospels. And they then exclude from teaching, and in some cases excommunicate (see 3 John) those who do not submit to these extra bells and whistles.
Why does John think that a teacher who refuses to teach what is in the
Gospels -theon ouch echei- does not have God? We could use the model
in the verses that follow John 3:16. "This is the -krisis-
that the light has come into the world, and people loved darkness rather
than light" (John 3:19). If the Gospels give us all we need to know
about the light of God, then anyone who rejects what is clearly taught
in the Gospels will be a lover of darkness, and therefore does not welcome
or have God in his or her heart.