I think that we in the Church have often lost a sense of the urgency of the Gospel. I'm not talking about heretics and schismatics, who might be expected to have lost this sense; I'm talking about even those of us who are orthodox, practicing Catholics, who believe everything that the Church and scriptures teach. Even among us, there is often a sort of lukewarm attitude, and it's born not from believing the wrong things, but about not necessarily taking the right things--which we do believe--seriously enough.
The consistent teaching of the Church, starting with Christ Himself, and continuing through the Church Fathers and throughout Church history, has been that there is only one narrow way to salvation: Through faith in Jesus Christ (and, by extension, the Holy Trinity--the God Who is Three in One--Whose Nature Christ revealed to us), baptism in the name of the Father, and of the Son, and of the Holy Spirit (as Christ taught one must be born again by water and the Spirit, not merely the Spirit), keeping Christ's commands, and repenting when we fail to do so. The Church and Christian history attests that faith in Christ includes faith in His Divinity as the Second Person of the Trinity, and faith in the miracles during Christ's life, including His birth of a virgin (it may seem secondary, but is in fact a confirmation that Christ is the begotten Son of God, not of a mortal father). Such is the nature of the beliefs and life that constitute the narrow way to salvation.
We have to be clear about something: The consequence of rejecting this narrow way, if the rejection lasts up through death, is the Second Death, an eternity in Hell. Let's think about that, let it really sink in. When we consider that people could die at any given time, it becomes serious business that they follow the narrow way. Hell is the absolute worst fate imaginable. Total separation from God's Grace, real pain--physical, spiritual, mental, emotional pain--worse than any we can imagine, and for all eternity! Do we really believe it?
If we really believed it and comprehended it, we would never be lax in our duty to evangelize. The very thought that someone even might be at higher risk of Hell would set our hearts aflame for their conversion and repentance! The sad truth, though, is that we often don't take it very seriously at all. We have it in our minds, I think, that damnation is rare. Whereas at one time Christians believed that it was necessary that a person not only believe in Jesus, but believe in the right things about Him and His Gospel ("If even an angel should preach to you a different Gospel, let him be anathema!" as St. Paul said), now we are very shy about saying these things are necessary. It's true that Christianity has held that God is not bound by His norms (although due to Scripture we know that even "exceptions" would somehow have to still be through Christ, as "no one comes to the Father but by Me"), so that we must never judge any individual soul as having gone to Hell, but we've taken a hope and turned it into a nigh certainty.
In fact, the only guarantee that someone will go to Heaven is that he holds to the narrow way of Christianity. Anything else would be a miracle of mercy. God can do that, but that possibility is not something we should assume. Anytime that others fail to believe in Christ, or to believe in all that He (and later the Holy Spirit) taught, we should be very passionate for them to amend their ways. We shouldn't be lukewarm. Such people are unambiguously outside of the normal means God has provided for salvation. God can save them anyway, if He so chooses (through mysterious means still owed to Christ's victory), but it's the same way that God could suspend someone in midair if he jumps off a cliff. We wouldn't fail to be gravely concerned and bothered if someone was going to jump off a cliff just because "God could send His angels to bear them up." No, we'd be overcome with the desire that the person choose the one known way to avoid falling to his death: To not jump or fall off the cliff in the first place. In exactly the same way, we should be passionate that people choose the one known way to avoid falling into Hell.
We should also know, from all this, that it's imperative that souls convert as soon as possible! It is true that we must meet people "where they are," but it should always be a matter of urgency that they convert, and fully, at the earliest possible time. The scriptures say that "today is the day of salvation." Not tomorrow. Not next month. Certainly not years from now. We are not promised a tomorrow. So the idea that sinners and unbelievers--including ourselves when we fall into such errors--have time to "gradually" convert is a dangerous and lethal notion. It could soften our urgency for their conversion. That's not to say we shouldn't accept that sometimes a person will only gradually repent or convert, nor that we mustn't be patient and charitable in our methods of appealing to them, but it means we must see to it that this gradual repentance is in spite of our urgent efforts, not because we had contented ourselves with their gradual conversion. After all, if we begin to believe that the unbeliever or unrepentant sinner has "plenty of time," then our passion for converting them will naturally suffer. We mortals tend to procrastinate if we think we have the luxury, so it's important we know that we don't.
I don't know how to kindle that passion, though. Even though I know the truth of what I say, my heart fails to grasp the gravity. It's as though the reality is so terrible, so burdensome, that my heart just cannot believe it. So I'm in the same boat as most people, when it comes to lacking a sense of urgency. For the most part, even if in my mind I know that "narrow is the gate and constricted the road that leads to life, and few are they who find it," in my heart I have the (false) sense of security that most people I know and encounter are probably going to go to Heaven regardless of their religious affiliation or lifestyle, because they seem like "good people" with "good intentions." I cannot fathom God sending or even allowing them to go to Hell for eternity. I confess that! Such is my limited human mind and heart that I cannot dare to grasp the mind and justice of God! That's my flaw, not any fault with God, Who is always without fault.
In the end, I believe the following standard is best: Judge as though everyone will go to Heaven, but evangelize as urgently as though the only souls in Heaven will be those who had explicit faith in Christ and His teachings. If we do that, we will be living the tightrope balance between "judge not" and "preach the gospel to all nations." And pray for me, a sinner, for I too have a long way to go.