We are commanded to repent apart from trusting in Him for redemption.
This mental conviction is a work. It is not baked into the belief of Jesus as Savior.
Faith is not the sole agency of redemption.
A dead faith is not as you have described a "empty profession of faith".
A dead faith is a faith without deeds.
Rahab had sincere faith in the God of the Hebrews but without her deeds her faith was dead.
Her faith was not a "empty profession of faith", it was real, it was sincere but it was not enough to save her.
In Romans 4:2-3, we read - For if Abraham was justified by works, he has something to boast about, but not before God. 3 For what does the Scripture say? “Abraham believed God, and it (faith, not works) was accounted to him for righteousness. Be sure to read Romans 4:2-6.
Now, do you believe that Abraham was saved when he believed God in Genesis 15:6 or not until many years later by works, after he offered up Isaac on the altar in Genesis 22? What do you believe "justified by works" in James 2:21 means? Accounted as righteous/saved by works or shown to be righteous by works?