There’s another section that applies here, in the SLBC 1689, chapter 5, paragraph 5:
The corruption of nature, during this life, doth remain in those that are regenerated; and although it be through Christ pardoned and mortified, yet both itself, and the first motions thereof, are truly and properly sin.
The Westminster has a similar paragraph, and adds another:
Every sin, both original and actual, being a transgression of the righteous law of God, and contrary thereunto, doth, in its own nature, bring guilt upon the sinner, whereby he is bound over to the wrath of God, and curse of the law, and so made subject to death, with all miseries spiritual, temporal, and eternal.
That pretty clearly teaches that the sin nature alone would condemn to hell. That said, I have leaned towards taking exception to this point along the lines of what @jander has suggested - holding that the sin nature received from Adam does not condemn us to hell in itself, but bends us inexorably to sin so that everyone who receives a sin nature sins. Regarding infants, I’ve thought that they too are guilty by reason of their own sin. While what does on the brain or consciousness of a tiny developing baby will always be a mystery to us, I would see them following into the sin with the first motion of their consciousness.
Romans 9:11 seems the biggest issue with this view, as they are told of the children in Rebekah’s womb, “For the children being not yet born, neither having done any good or evil…” I’m not sure how to reconcile this with other verses which talk about everyone being judged for their deeds, perhaps Romans isn’t excluding sins of omission (like not worshiping and submitting to God in the mind,) or it’s speaking about actions rather than thoughts.
I’m not settled in these positions, and I appreciate the discussion.