The JST retains the more difficult text, but attempts to make its import clearer through the additional text. The earlier 3 Nephi solution was simply to replace it.
Of course, the ultimate meaning of both the New Testament and 3 Nephi passages is that we must root out evils that might damage our soul. Sometimes they are very important to us; they may actually be physical addictions. Many a convert to the Church understands the difficulty of overcoming Word of Wisdom addictions. The same is true of mental/emotional habits. The harm to our souls comes from the long-term harboring of feelings that may escalate to more damaging feelings or actions. We are to remove them. The Matthean metaphor is cutting off an important part of the body. The 3 Nephi example is taking up the cross, a symbol of how the atonement purifies us of sin.