We said the same thing except you left out the phrase 'all their sins', so I reincluded it.
Apparently, I'm going to have to explain this more slowly.
You wrote--
The absence of 'saving faith' is the sin that damns.
and I corrected that statement to "No, it's the absence of faith."
So what's the difference between the two? It's just this...
The absence of Faith is, in itself, what would deny someone salvation, NOT the commission of a sin. Not even what you identified as the sin of unbelief.
In other words, it's not because of the sin (of unbelief) but of the lack of saving Faith.
We all sin, even after coming to Christ! And all of them are subject to being forgiven by God, but that applies only to those people who have Faith in the saving work of Christ (who told his followers not to worry about their salvation so long as they do trust in Him).