That doctrine that you summarize formed a big turning in my understanding. I grew up in independent Bible churches and Baptist churches, both of which teach the doctrine that salvation by Christ is primarily the payment for our sin debt, and that we accept that payment by faith, but that we become forgiven sinners that should sin less and less as we grow on Christ (progressive sanctification). Further, this Baptist doctrine teaches that we will continue to sin to some extent, until we leave this earthly life and become instantly free of sin.
There are a whole set of "explanations" developed to interpret those verses that you refer to, such as positional truth categorizing that says that those verses describe how God sees you in your position as a child of God, through the "filter" of Chrisi's blood, not how you actually are.
On the other hand, there are other doctrinal systems that take those verses as present reality. The Wesleyan and many Pentecostal groups invented a second work of grace doctrine that says that after receiving the pardon for our sins, we are expected to reach a stage in which we achieve sinless perfection through a complete dedication to the Holy Spirit. Details of how this is done vary in the different groups, along with how much time this takes and at what point of failing to become sanctified or to maintain sanctification, a pardoned sinner might lose his salvation.
These two doctrinal lines are in competition so if one group hears you say something, even just a Bible quote, which echoes a phrase from the opposing doctrine, you will usually get a challenging accusation of supporting and teaching THAT doctrine.
About 45 years ago, I took a challenge seriously to accept the Bible for what it says, without the manufactured doctrines. At first, it seemed impossible. But God had mercy on my ignorance and opened my eyes to see that the Bible speaks in 2 realms, the spirit and the flesh. That resolves the confusion, because it is our inner person that is born again, not our body of flesh. Both groups are partly right and both are very wrong.