"Christians believe either that Christ's righteousness is accounted as the Christian's at the time of judgment (Protestant) or that Christ's righteousness actually makes the Christian righteous through sacraments and good deeds (Catholic)."
That is exactly what I wanted you to say.
Let's examine the logic behind that statement. Not to say that I don't believe it because after all it is completely true.
If someone belongs to a group, (lets say Catholics) as Gods creations why are his good deeds good enough to get him into heaven, whereas an equal creation of God (a Protestant) will not enter the kingdom under these terms. The Protestant must repent. That makes no sense.
Actually, from this Jew's perspective it means that, one way or another, we're all going to need a lot of forgiveness. G-d, thankfully, is more than adequate in that department.
No, it doesn't make sense. That's because you are coming at it through a faulty premise (strawman). Catholics don't believe we will get into Heaven because of our good works. We believe, as all Christians believe, we will get to Heaven because of Christ's sacrifice. However, we are still human. We discipline the flesh to the will of the Father through our works because the the flesh is weak.
Read the Book of James... a book largely ignored by most Protestants I encounter. We are not justified by our works but by our faith. We have an empty faith, though, if it doesn't include works.
All of this assumes, of course, that our ultimate fate depends on our opinions on these matters, more than does God's opinion about us.
Once you toss God's opinion into the mix, none of that particular controversy makes a lot of sense.