Why aren't all unbelievers given a "Road to Damascus" moment, according to Catholicism?

TL;DR (1) Because He doesn't need to, and (2) it's not His style.

1 Samuel 3:18

What is pleasing in the LORD’s sight, the LORD will do.

If you look at how well the Apostle Paul spread the Faith, particularly among the gentiles, how he (and all of the Apostles and Church Fathers) spread the Good News (of Jesus), and if you consider that the Faith remains alive and well 2000 years later then God chose his servant(s) well. The Good News is now available all over the world, for those who will listen, which it wasn't when Paul had that experience on the way to Damascus.

Demanding that one receive a special message identical to Paul's from God is to deeply misunderstand the relationship between man and God, and in so doing step smartly into the deadly sin of pride.

Paul didn't demand this message; the message Paul got was a gift from God (possibly "undeserved" given Paul's previous opposition to Christians) that fulfilled God's purpose.

1996 Grace is favor, the free and undeserved help that God gives us to respond to his call to become children of God, adoptive sons, partakers of the divine nature and of eternal life.

If you, or I, or an unbeliever tries to demand a particular form of grace from Him ... good luck with that. What your question amounts to is: "If I were God, I'd have done it this way..."
You're not God and God does it His way. We are called to open our hearts and trust God. Doubting Thomas was chastised for having the wrong attitude. (See John 20:25-29, below).

Catechism Reference on the relationship between God and Mankind

CCC 42 God transcends all creatures. We must therefore continually purify our language of everything in it that is limited, image-bound or imperfect, if we are not to confuse our image of God--"the inexpressible, the incomprehensible, the invisible, the ungraspable¹"--with our human representations. Our human words always fall short of the mystery of God. 43 Admittedly, in speaking about God like this, our language is using human modes of expression; nevertheless it really does attain to God himself, though unable to express him in his infinite simplicity. Likewise, we must recall that "between Creator and creature no similitude can be expressed without implying an even greater dissimilitude"; and that "concerning God, we cannot grasp what he is, but only what he is not, and how other beings stand in relation to him."

God communicated with Saul of Tarsus (Paul) in a particular way for His own reasons. God will speak to others in other ways to fulfill His purposes. Whether or not we are open to, or will listen to, the message is a different matter. Pride often gets in the way of listening to God.

God's method, shown in Scripture, is to pick certain servants to get His word out to others

There's a whole world full of Christians (Catholic and otherwise) striving to get the Good News of Jesus to the non-believer. The Great Commission called to the faithful during the Apostolic age, and 2000 years later still calls all Christians to do likewise(CCC 1-3). Even before the Apostolic age God called certain individuals to go forth and get His words out to others: the prophets. The Old Testament shows that God's "style" is to pick someone to go forth and do what He needs done.

Entire books of the bible are devoted to the prophets. One example (often used in homilies and teachings by Catholic clergy) is Samuel (From 1 Samuel 3)

9... When Samuel went to sleep in his place, 10 the LORD came and stood there, calling out as before: Samuel, Samuel! Samuel answered, “Speak, for your servant is listening.” 11 The LORD said to Samuel: I am about to do something in Israel that will make the ears of everyone who hears it ring. 12 On that day I will carry out against Eli everything I have said about his house, beginning to end. 13 I announce to him that I am condemning his house once and for all, because of this crime: though he knew his sons were blaspheming God, he did not reprove them. 14 Therefore, I swear to Eli’s house: No sacrifice or offering will ever expiate its crime.* 15 Samuel then slept until morning, when he got up early and opened the doors of the temple of the LORD. He was afraid to tell Eli the vision, 16 but Eli called to him, “Samuel, my son!” He replied, “Here I am.” 17 Then Eli asked, “What did he say to you? Hide nothing from me! May God do thus to you, and more,* if you hide from me a single thing he told you.” 18 So Samuel told him everything, and held nothing back. Eli answered, “It is the LORD. What is pleasing in the LORD’s sight, the LORD will do.”

Samuel listened. Samuel was open to God's word. So have been people of all sorts for the past two thousand years. (The list of Christian mystics is too long to get into here ...)

For a more recent example, Saint Francis of Assisi didn't require a Pauline experience to go forth, change his life, and then do God's work. Why does anyone else need or require it?

Why didn't He give you, or me, or an unbeliever Paul's experience?

Two ways to answer that question.

1. You and I don't know that he didn't and that someone completely missed the message. A variety of people over the years have reported profound spiritual experiences, some of whom are mystics, others of whom have shared or experienced a more basic conversion experience. How do you or I know that such an experience wasn't the divine message (like the message to Paul) for that person? We don't know.

   The core of the answer you've already touched on: Jesus explains the call to believe, even if you haven't seen, in John 20: 25-29 via doubting Thomas.

   25 So the other disciples said to him, “We have seen the Lord.” But he said to them, “Unless I see the mark of the nails in his hands and put my finger into the nail marks and put my hand into his side, I will not believe.” 26 Now a week later his disciples were again inside and Thomas was with them. Jesus came, although the doors were locked, and stood in their midst and said, “Peace be with you.” 27 Then he said to Thomas, “Put your finger here and see my hands, and bring your hand and put it into my side, and do not be unbelieving, but believe.” 28 Thomas answered and said to him, “My Lord and my God!” 29 Jesus said to him, “Have you come to believe because you have seen me? Blessed are those who have not seen and have believed.”

2. He didn't need to. He's provided the entire Church (in terms of all of the Faithful, the entire (mystical) Body of Christ) to get His word out and to witness to Him. All of those who have answered the call to Faith, to be in a state of communion with God (CCC 1-10) are like Samuel (or should be). Speak Lord, for your servant is listening.

   Beyond the Great Commission in Matthew -- which the Catholic Church accepts regardless of secular criticisms of that passage in Scripture -- the spreading of the Good News is the spreading of God's word. Going forth from the Apostolic age, anyone receiving the Good News is asked to make the leap of Faith, to trust God, to believe. The vignette about doubting Thomas emphasizes this, and it should be given due weight given that Jesus is the one issuing the rebuke to Thomas ... and through Thomas to any who doubt.

Graces given repaid by the sins of pride and envy

Paul was given a particular Grace, a Gift from God, that took a particular form. Others are given graces that take other forms. Among the graces given to us is the people that we know and meet who bring God's word to us, in both word and deed. If we reject them -- nope, I'm not buying it until Jesus strikes me blind for three days -- how do we know that we have not just rejected God's grace? We don't.

• Parent moment: "You gave Johnny that, why didn't you give me that?" What's that parent's answer? Because that's what Johnny needed, and you'll get something else that you need.

Issuing Ultimatums to God

The question as asked infers a particular attitude vis a vis the relationship between man and God.

I won't believe unless you give me a message the way you gave Paul the message

The sin of Pride fairly leaps off of the page.

Charity 2093 Faith in God's love encompasses the call and the obligation to respond with sincere love to divine charity. The first commandment enjoins us to love God above everything and all creatures for him and because of him.

CCC 2094 One can sin against God's love in various ways:
- indifference neglects or refuses to reflect on divine charity; it fails to consider its prevenient goodness and denies its power. - ingratitude fails or refuses to acknowledge divine charity and to return him love for love. - lukewarmness is hesitation or negligence in responding to divine love; it can imply refusal to give oneself over to the prompting of charity. - acedia or spiritual sloth goes so far as to refuse the joy that comes from God and to be repelled by divine goodness. - hatred of God comes from pride. It is contrary to love of God, whose goodness it denies, and whom it presumes to curse as the one who forbids sins and inflicts punishments.

CCC 1866 Vices can be classified according to the virtues they oppose, or also be linked to the capital sins which Christian experience has distinguished, following St. John Cassian and St. Gregory the Great. They are called "capital" because they engender other sins, other vices. They are pride, avarice, envy, wrath, lust, gluttony, and sloth or acedia.

The willful decision to ignore the message sent via doubting Thomas in Scripture -- Scripture is God's word to us -- via the Faith handed down to us from the Apostolic age to now = Pride and Arrogance.

• Who are you, or who am I, or who is an unbeliever to tell God how to talk to speak to us? He speaks to us in His own time and choosing, for His own purposes. We are called to have faith, and to trust in Him.
  Jeremiah 29:11

  For I know the plans I have for you,” declares the LORD, “plans to prosper you and not to harm you, plans to give you hope and a future."

Why didn't I get the same gift/message Paul got?

Because He has other gifts for you, manifestations of His grace.

The seven capital sins:
1. Pride ; 2. Covetousness; 3. Lust; 4. Anger; 5. Gluttony; 6. **Envy; 7. Sloth

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Aside: the question is lacking in form

The "why didn't someone do something" question form is difficult enough when we are dealing with humans. It typically requires making a guess or an inference, and is often unknowable. (Personal experience from fatal aircraft accident investigations: why didn't (the pilot) do X?).

When dealing with the Almighty, it's even more difficult to discern if that is even possible. (See above, CCC ref, about God being "ingraspable" and otherwise mysterious to humans).

You can sometimes work your way back to "why did someone do something?" and so we learn that God did send Paul the message that way for a particular purpose: to get him to apply his considerable energy and talent for Christ. It worked. God did pass his instructions to Samuel (and all of the other prophets ...)

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¹ Liturgy of St. John Chrysostom, Anaphora.