That then also means a born again believer can remain in a state of non-belief,
Yes, they remain in unbelief as concerning the doctrine that Jesus taught, however. they do believe that God is their heavenly Father and they serve him the best that they can with their misinterpretation of the doctrine.
The only way to that anyone has knowledge of the doctrine that Jesus taught is from the revelation of the Holy Spirit within them. I found in my experience that as long as you think you are intelligent enough to figure out the truths that are contained in the doctrine, the Holy Spirit within you will not reveal the understanding of those truths. Otherwise, you will take the credit instead of giving it to God.
I grew up going to church with my parents, but never retained much knowledge of the scriptures. Every time I tried to make sense of the scriptures they seemed to contradict each other, mostly on the salvation scriptures. I attended all kinds of different churches, taking notes. When I turned 50 years old, I bought a concordance hoping to understand better, with no avail. I was stubborn enough that I studied diligently for 12 years and finally told myself that I was not smart enough to figure it out. Within a very short time, gradually, the contradicting scriptures began to harmonize
When I realised this, the scripture Mark 8:34 came to mind, Whosoever will come after me, let him deny himself, and take up his cross and follow me.
I believe many students of the bible, who are intelectual, have trouble such as this, and are believing their eternal deliverance is due to their good works.