By your definition a Christian is a 'born again' but that unpleasant dogma is NOT mainstream.
I think it is mainstream, but what the interpretation of what it means to be born again , varies.
Some churches just see it as meaning you attend a church and become and live life as a Christian others think it means something more.
So Christians I have known sometimes don't see the C of E members as having been "born again" because they have a different angle on it.
Born again, can then become exclusive, with people saying things like the c of e is dead etc.
But even the c of e has a concept of being born again, they just don't expect members to start speaking in tongues to prove they have it.
That's what I have found.
being born again is in the bible it's just the interpretations of what that means that changes.
The c of e doesn't expect its members to speak in tongues and doesn't see it as separate to living a church life, other even mainstream Christians ( Pentecostal) sometimes don't seem to see the members of c of e as very spiritual or born again.
It's a bit snobby, thinking other Christians arn't " born again' because they don't worship or speak in tongues in the same way.
But the idea of being born again is mainstream it's just the c of e version is more " sensible" imo