That begs the question, "What did he enter our hearts to do?' I believe that he has come into our hearts to make us followers of his (and the Apostle's) teachings.
This is the right question because it gets to the heart of the matter. Jesus said He (and His Father) would take up residence in our hearts in order to make Himself known to us, to lead us, to guide us, to teach us, to warn us, to correct us, to comfort us, and to make us one with Him and the Father (John 14-17). With Jesus in our hearts, we are complete (Col 2:9-10), we have everything we need for life and godliness (2 Pet 1:3), and we possess His life, yes His eternal life (1 Jn 5:11-12).
To follow a way of life (I have heard that Christians were first called "people of the way.") IMO a "way" of going or indicating that one is taking the actions up of the people who teach how to live the "way." who teach not just what to think but what to do to stay on that path. As we all know "narrow is the and difficult is the way." I take something like that and "take up your cross" as a call to action.
Jesus is the gate and He is the way. It is unfortunate that some tranlations characterize the way as "difficult". NASB, NIV, and YLT more appropriatly capture the meaning.
13 Enter through the narrow gate; for the gate is wide and the way is broad that leads to destruction, and there are many who enter through it.
14 For the gate is small and the way is narrow that leads to life, and there are few who find it. (Mt 7:13–14 NASB 1995)
13 “Enter through the narrow gate. For wide is the gate and broad is the road that leads to destruction, and many enter through it. 14 But small is the gate and narrow the road that leads to life, and only a few find it. (Mt 7:13–14 NIV)
13 ‘Go ye in through the strait gate, because wide is the gate, and broad the way that is leading to the destruction, and many are those going in through it; 14 how strait is the gate, and compressed the way that is leading to the life, and few are those finding it! (Mt 7:13–14 YLT)
Jesus made it obvious that the way is not difficult:
28 Come to Me, all you who labor and are heavy laden, and I will give you rest. 29 Take My yoke upon you and learn from Me, for I am gentle and lowly in heart, and you will find rest for your souls. 30 For My yoke is easy and My burden is light. (Mt 11:28–30)
But those who labor and are heavy laden don't see it that way, for obvious reasons.
Once he does take up residence in our hearts, and forgives us of our sins. As a child of God through spiritual birth, speaking for myself, I would want to make him (our Father) proud by following his example and instructions. For me, it wouldn't do to just acknowledge that he is my father without following his instructions.
It may be semantics only, as you mention below, but I do not have such a distant relationship with the Father. I do not try to please Him by "following His example (when He lived on the earth) and (His written) instructions". He lives in my heart. He tells me what to think, where to go, what to do, what to say, and how to say it. When I do not follow His instructions, He corrects me. We have an intimate personal relationship.
Very well put "Walk where he leads." As a Christian, my experience with the few Churches that I have been involved with, conservative, liberal, Catholic, and Protestant, has been a consistent message in all of the Churches I know. It's not just about agreeing that the walk is right, it's more about taking that walk or taking up our cross. It's difficult and we are not born perfect. That is why when we falter on our walk, we are forgiven.
I do recognize that though you state your affiliation as "Catholic", you seem have a broader view of Christianity.
I don't know if it's just semantics but it's not that we are favored due to our achievements or "works" as it were. It's that works are not the cause of our faith but our works (how we live and treat other people) are the evidence of our faith.
This is part of Baptist doctrine also. But it has its flaws. One main flaw is that if good works are evidence of salvation, then what is sin evident of? The most compelling answer is that if sin is followed by repentence then repentance is evidence of salvation. To me, this loophole destroys the original claim. The better paradigm is to say that whatever good is in me is because Jesus lives in me.
St James wrote an entire book on this topic which is often ignored in order to favor a portion of St Paul's writings. Which do conflict in text but in interpretation.
I wouldn't say that the entire Book of James discusses a single topic. Effectivness of prayer and the lifecycle of sin are two examples of other topics. But I do think you are right that James' statements about faith without works being dead are often ignored by those who don't believe in faith plus works. But proponents of faith plus works also ignore some things James says.
For example, James says,
10 For whoever shall keep the whole law, and yet stumble in one point, he is guilty of all. 11 For He who said, “Do not commit adultery,” also said, “Do not murder.” Now if you do not commit adultery, but you do murder, you have become a transgressor of the law. 12 So speak and so do as those who will be judged by the law of liberty. 13 For judgment is without mercy to the one who has shown no mercy. Mercy triumphs over judgment. (Jas 2:10–13 NKJV)
The above paragraph leads into James' discussion that faith without works is dead. What does it mean? It means if someone says they are dead to the law through the body of Christ (Ro 7:4) but are judgmental towards other people who break the law, their "faith" is innefective. If a person believes they are not under law but under grace, then they should "speak and do" (i.e., "act") "as those who will be judged by the law of liberty".
