teaching acknowledged that a person must continue to believe, or else they were never saved to begin with
This is the Biblical teaching, except, that a person "will" continue to believe, else they were not converted. "Must", as if ongoing salvation depended on ongoing believe, is fully the wrong idea.
You introduce your doubt of the surety of rebirth, where the real question is concerning the authenticity of rebirth. And the Bible tells us that God affirms to us that we are His. He has affirmed that to me and I am certain. And why cast aside the certainty of His salvation, thinking that somehow I might by my doing this or not doing that I'm going to toss it away?
What kind of salvation is that?
God assures us over and over, addressing every objection, if you are willing to just take it all for exactly what it says, and not settling for less.
And one of the most heartfelt concerns we have, our ability to live pure lives, is itself addressed in the knowledge of the certainty of our salvation. He who has this hope purifies himself, even as He is pure.
Jesus is made unto us all that we need, and He will not fail us.
Much love!