I have a few questions. This happens to be a huge atheist/Christian debate topic, mostly because Christians have many flavors of repentance.
A. The main one: Does repentance matter or is faith alone all that is needed? Take the case of a faithful individual who doesn't acknowledge an error. Does it matter? Does someone who is already "saved" really need Jesus Christ to keep on forgiving him?
Repentance most definitely matters (in that case and any other). Salvation is achieved by God's grace alone (God's unmerited favor towards humanity), through one's faith alone (one's trusting in Jesus Christ and his finished work -- namely his perfect life, death, and resurrection). One's personal effort ("doing good works") does nothing in terms of receiving salvation.
*** But repentance isn't necessary then, right?? (because repentance is one's personal effort, right??) ***
Repentance (in an overall sense, rather than solely an interpersonal sense) involves a "transformed mind" with regard to sin (a change in how one perceives sin). Repentance (overall) isn't something that can be accomplished solely on one's own. It is also a work of God (see Psalm 51 for The Bible's prime example of repentance). However, it is up to the individual to actively seek the ways of God rather than remaining stagnant within one's own sinful ways. Into The Night did a very good job of conveying the interpersonal aspect of repentance, providing "steps of repentance" that individuals should take whenever they've wronged someone.
Continued repentance (overall) involves a state of mind in which one continually wars against one's own sinful nature (one's own natural tendency to commit sin) rather than making friends with that sinful nature. Thus, repentance (turning away from sin) is something that goes hand in hand with the salvation that is received through grace by faith (turning towards God).
God has already done the "heavy lifting" (via Jesus living a perfect life, dying on the cross, and then rising from the dead). You need only have faith in his work.
B. Jesus died to save everyone; is He really going to go through all that just to not forgive someone a trespass?
Yes, because some people refuse to receive God's gift of salvation.
C. Atheists and Protestants agree on your second point, but Catholics believe that Contrition, i.e. "confessing to a priest" absolves one of his sins. The wronged party is never faced or compensated because God has already done the forgiving, which is all that matters in Catholicism. (In Catholicism, one's soul is like one's clothing; it needs to be laundered periodically, which is what Contrition accomplishes, and one is free to go as long as one wants without laundering one's clothes, but he will be judged by his attire at the moment his time comes and the bouncer in the sky decides whether or not he gets into the club).
Catholicism has this wrong. The wronged party should absolutely be faced (in addition to facing God). Both Matthew 5 and Matthew 18 make this quite clear.
I see that Into The Night noted that "the Protestants aren't much better". He's correct about that (in a generalized sense, which is all that one can really speak of with regard to so many people, unless one wishes to nuance the heck out of each and every post made on this forum), but I'll just make a quick note for you that there are in fact a minority of "Protestants" who are much more biblical in their views (who do NOT encourage woke policies or attempt to change Jesus into a woman), such as the Wisconsin Evangelical Lutheran Synod" (WELS) church that I am a member of (and regularly attend).
The general term "Protestant" encompasses a whole plethora of different "church denominations" with all sorts of different viewpoints about all sorts of different particulars. Then, there's the fact that even more specific terminology within Protestantism, such as "Baptist" or "Lutheran", still encompasses a plethora of different "church denominations" with all sorts of different viewpoints about all sorts of different particulars. "Protestantism" in general is just one big hosh posh of a mess that can distract away from what truly matters, which is "The Plan™".
D. Are the five items you list above an algorithm or can they be done in a different order, e.g. forgive oneself first and jump out of the block of code?
I think the order that ITN listed them in is pretty sensible (but not a fixed block of code).
E. Your last item in the list above indicates that one must follow the promptings and teachings of Jesus Christ, i.e. be a good person and live a good life, ... but that maxim seems to immediately get tossed out the 1-cubit x 1-cubit window every time, and replaced with "faith is all that matters", i.e. it's only a question of what one believes, which totally nullifies that final item on your list.
He specified "for the Christian" on that one, otherwise I do see where you're coming from here.