The Catholic Church seems to teach that we cannot ask the saints/angels for anything else other than to pray for us, but I don't understand why?

The Catholic Church seems to teach that we cannot ask the saints/angels for anything else other than to pray for us, but I don't understand why?

It is true that in general, we may ask the Holy Angels or saints in heaven to pray for us. But that does not mean that they do not intervene in our earthly life in other ways. Miracles asked for are accomplished through God’s power, that much has never been doubted by the Church.

Healings from the Holy Angels and Saints may occasionally happen, but they are somewhat rarely brought about.

How much devotion we have to a particular Angel or saint may influence how often we get answers to our prayers. Also, we should try to meet God halfway and do our part. Personal sanctification and holy need to be done earnestly on our part.

Personal devotion to a particular saint or Holy Angel can be rewarded without even asking for help or physical healing.

Sr. Lúcia de Jesus Rosa dos Santos once asked the Virgin Mary why so few people ever get healed. Our Lady’s response will never leave my mind: ”Because God is leery of men and He knows their hearts.”

Catholic Answers explains quite well why Catholics pray to Saints:

Another charge commonly levelled against asking the saints for their intercession is that this violates the sole mediatorship of Christ, which Paul discusses: “For there is one God, and there is one mediator between God and men, the man Christ Jesus” (1 Tim. 2:5).

But asking one person to pray for you in no way violates Christ’s mediatorship, as can be seen from considering the way in which Christ is a mediator. First, Christ is a unique mediator between man and God because he is the only person who is both God and man. He is the only bridge between the two, the only God-man. But that role as mediator is not compromised in the least by the fact that others intercede for us. Furthermore, Christ is a unique mediator between God and man because he is the mediator of the New Covenant (Heb. 9:15, 12:24), just as Moses was the mediator (Greek mesitas) of the Old Covenant (Gal. 3:19–20).

The intercession of fellow Christians—which is what the saints in heaven are—also clearly does not interfere with Christ’s unique mediatorship because in the four verses immediately preceding 1 Timothy 2:5, Paul says that Christians should intercede: “First of all, then, I urge that supplications, prayers, intercessions, and thanksgivings be made for all men, for kings and all who are in high positions, that we may lead a quiet and peaceable life, godly and respectful in every way. This is good, and pleasing to God our Savior, who desires all men to be saved and to come to the knowledge of the truth” (1 Tim. 2:1–4). Clearly, then, intercessory prayers offered by Christians on behalf of others is something “good and pleasing to God,” not something infringing on Christ’s role as mediator.

Not only do those in heaven pray with us, they also pray for us. In the book of Revelation, we read: “[An] angel came and stood at the altar [in heaven] with a golden censer; and he was given much incense to mingle with the prayers of all the saints upon the golden altar before the throne; and the smoke of the incense rose with the prayers of the saints from the hand of the angel before God” (Rev. 8:3-4).

And those in heaven who offer to God our prayers aren’t just angels, but humans as well. John sees that “the twenty-four elders [the leaders of the people of God in heaven] fell down before the Lamb, each holding a harp, and with golden bowls full of incense, which are the prayers of the saints” (Rev. 5:8). The simple fact is, as this passage shows: The saints in heaven offer to God the prayers of the saints on earth. - Praying to the Saints

The Catechism of the Catholic Church has this to say on the Communion of Saints of heaven and earth.

II. The Communion of the Church of Heaven and the Earth

954 The three states of the Church. "When the Lord comes in glory, and all his angels with him, death will be no more and all things will be subject to him. But at the present time some of his disciples are pilgrims on earth. Others have died and are being purified, while still others are in glory, contemplating 'in full light, God himself triune and one, exactly as he is"':

All of us, however, in varying degrees and in different ways share in the same charity towards God and our neighbours, and we all sing the one hymn of glory to our God. All, indeed, who are of Christ and who have his Spirit form one Church and in Christ cleave together.

955 "So it is that the union of the wayfarers with the brethren who sleep in the peace of Christ is in no way interrupted, but on the contrary, according to the constant faith of the Church, this union is reinforced by an exchange of spiritual goods."

956 The intercession of the saints. "Being more closely united to Christ, those who dwell in heaven fix the whole Church more firmly in holiness.... [T]hey do not cease to intercede with the Father for us, as they proffer the merits which they acquired on earth through the one mediator between God and men, Christ Jesus.... So by their fraternal concern is our weakness greatly helped."

Do not weep, for I shall be more useful to you after my death and I shall help you then more effectively than during my life.

I want to spend my heaven in doing good on earth.

957 Communion with the saints. "It is not merely by the title of example that we cherish the memory of those in heaven; we seek, rather, that by this devotion to the exercise of fraternal charity the union of the whole Church in the Spirit may be strengthened. Exactly as Christian communion among our fellow pilgrims brings us closer to Christ, so our communion with the saints joins us to Christ, from whom as from its fountain and head issues all grace, and the life of the People of God itself":

We worship Christ as God's Son; we love the martyrs as the Lord's disciples and imitators, and rightly so because of their matchless devotion towards their king and master. May we also be their companions and fellow disciples!

958 Communion with the dead. "In full consciousness of this communion of the whole Mystical Body of Jesus Christ, the Church in its pilgrim members, from the very earliest days of the Christian religion, has honored with great respect the memory of the dead; and 'because it is a holy and a wholesome thought to pray for the dead that they may be loosed from their sins' she offers her suffrages for them." Our prayer for them is capable not only of helping them, but also of making their intercession for us effective.

959 In the one family of God. "For if we continue to love one another and to join in praising the Most Holy Trinity - all of us who are sons of God and form one family in Christ - we will be faithful to the deepest vocation of the Church."

St. Mary of Jesus Crucified had her throat cut and was miraculously healed by the Virgin Mary, in whom she had a great devotion. Mary saved her life and nurtured Miriam back to health without her even asking to be cured. Mary saved her most surely due to her continuous devotion. She bore the scars of this attack the whole of her life.

On September 8, 1858, Mariam convinced him she would never abandon her faith; in response, he cut her throat and dumped her in an alley. An apparition of the Virgin Mary treated her wound and she lived. Her description of the Virgin Mary’s apparition, A nun dressed in blue picked me up and stitched my throat wound. This happened in a grotto somewhere. I then found myself in heaven with the Blessed Virgin, the angels and the saints. They treated me with great kindness. In their company were my parents. I saw the brilliant throne of the Most Holy Trinity and Jesus Christ in His humanity. There was no sun, no lamp, but everything was bright with light. Someone spoke to me. They said that I was a virgin, but that my book was not finished. ”

She then found herself once again in the grotto with the “nun dressed in blue.” How long did Mariam remain in this secret shelter? She later spoke of one month, but she was not sure. One day, the unknown nurse prepared some soup for her that was so delicious that she greedily asked for more, and all her life she was to remember the taste of this heavenly soup. On her death bed she was heard to say tenderly, “She made me some soup! Oh, such good soup! There I was a long time, looking, and never ate soup like that. I have the taste in my mouth. She promised me that at my last hour, she will give me a little spoonful of it.” - St. Mary of Jesus Crucified

Well, for the most part, it may be seemingly true that the Angels, especially our Guardian Angels and saints, in particular our patron saints or those saints that we have a particular devotion to simply just intercede on our behalf, but they occasionally are known to do much more. Most of their actions will naturally go unnoticed.

Over the years, I have come across many stories of how saints and angels have helped people in very concrete ways.

I will start with a story of a friend, who had a devotion to Saint Joseph and as a little girl got lost, during WWII and finally realized she was in the middle of a mine field. In panic, she prayed to St. Joseph. Eventually a man showed up and showed her not only how to get out of the mine field, but also how to get home. In the end, the complete stranger simply disappeared. My friend is still very much alive and truly believes the man was St. Joseph.

The Sisters of Loretto, Santa Fe, New Mexico, believe St. Joseph built the miraculous spiral staircase in their chapel, after praying to St. Joseph for help.

The person who built the staircase said he would accept his wages when he had finished his work. However, he disappeared when the staircase was complete and the Sisters believe it was St. Joseph who did this miraculous work.

Loretto Chapel is best known for its "miraculous" spiral staircase, which rises 20 feet (6.1 m) to the choir loft while making two full turns, all without the support of a newel or central pole. The staircase is built mostly out of wood and is held together by wooden pegs, with no glue, nails or other hardware used. The inner stringer consists of seven wooden segments joined together with pegs, while the longer outer stringer has nine segments. The exact wood used to build the staircase has been confirmed to be a type of spruce which is not native to New Mexico and scientifically not identified anywhere else in the world.

The handrails were added later in 1887, and an iron bracket was later attached to a column to add additional support. The staircase is supported by an inner wood stringer.

Apart from any claims of its miraculous nature, the staircase has been described as a remarkable feat of woodworking. According to a Washington Post column by Tim Carter:

"It's a magnificent work of art that humbles me as a master carpenter. To create a staircase like this using modern tools would be a feat. It's mind-boggling to think about constructing such a marvel with crude hand tools, no electricity and minimal resources."

According to another professional carpenter, who was interviewed by Ben Radford for his book Mysterious New Mexico:

"The execution is just incredible. The theory of how to do it, to bend it around in a two-turn spiral, that's some difficult arithmetic there." - Loretto Chapel

St. Frances of Rome enjoyed the privilege of being able to see her Guardian Angel.

We read in the life of St. Frances of Rome that during the latter half of her life she enjoyed the singular grace of seeing her Guardian Angel ever at her side. She thus describes this heavenly companion: "His aspect is full of sweetness and majesty. His eyes are generally tumed toward Heaven, and words cannot describe the Divine purity of that gaze. His brow is always serene; his glances kindle in the soul the flames of ardent devotion. When I look upon him, I understand the glory of the angelic nature and the degraded condition of our own." Her wish had always been to attain a perfect conformity with the Divine will, and now this mysterious guidance furnished her with the means of knowing that will in its minutest details. In her struggles with the evil one, the Angel became her shield of defense.

The presence of her heavenly Guide was also to Frances a mirror in which she could see reflected all her imperfections. When she committed the slightest fault, the Angel disappeared; and it was only after she had carefully examined her conscience, discovered her failing, lamented and humbly confessed it, that he returned. On the other hand, when she was only disturbed by a doubt or a scruple, he dissipated her uneasiness by a look of great kindness. His guidance enlightened her chiefly with regard to the difficulty she had in submitting to certain cares and obligations which belonged to her position as mistress of a family. She was wont to imagine that the hours thus employed were lost in God's sight, but the celestial Guardian corrected her judgment on this point and taught her to discern the Divine will in every little irksome duty, in every contradiction, great or small, as well as in great trials and on important occasions. - St. Frances of Rome

One day, St. Catherine Labore was awakened by her Guardian Angel in the form of a child who led her to the chapel to converse with the Blessed Virgin Mary, then led her back to her room. How cool is that!

You asked:

If it's not a sin here on Earth to ask a doctor to heal me, instead of asking him to pray to God to heal me, why can't I do something similar with an angel?

which is to assert that when you are healed under the care of a doctor that the healing wasn't in accord with the active will of God, but only His passive will.

Imagine rolling down a mountain and having one boulder pops your your shoulder out of its socket only to have the next one set it into place.

But if it is a miracle, you could say

Objection 4. Further, the miraculous restoring to health is done by the power of God. Therefore the grace of healing should not be distinguished from the working of miracles.

but the purpose of the miracle that is important

Reply to Objection 4. The "grace of healing" is mentioned separately, because by its means a benefit, namely bodily health, is conferred on man in addition to the common benefit bestowed in all miracles, namely the bringing of men to the knowledge of God.

https://www.newadvent.org/summa/3178.htm

and there are other miracles that you can ask for - signs, grace etc...

But the fundamental truth is that all things are done in accord with God's will. But there has to be some benefit to knocking and some benefit to persistence, which is clearly taught by Jesus when he coined the phrase "the squeaky wheel gets the grace". Luke 11:5-8

so it's not that you can't ask a Saint or angel to do anything more than pray for you, but whatever you ask a Saint or angel must be :

1. Miraculous
2. to further God's glory
3. Good
4. Under God's authority.
5. In accord with God's active will.

Angels are, quite literally, messengers of God. All their power is through Him. Venerating them is in order; adoring them, worshiping them is not: That would be idolatry. Praying to a saint or an angel and asking them to perform divine miracles on their own would make them God-like. Asking them to intercede, to pray for us, by contrast, is something we can ask any Christian, dead or alive or immortal, in the hope that God hears them.¹

This website explains the issue quite nicely. The relevant quote from the Bible is probably Revelation 19:10 (emphasis by me). The prophet of the Book of Revelation starts to worship an angel; the angel gently and humbly corrects him, saying that he is not different from the narrator himself:

9 Then the angel said to me, “Write this: Blessed are those who are invited to the wedding supper of the Lamb!” And he added, “These are the true words of God.”

10 At this I fell at his feet to worship him. But he said to me, “Don’t do that! I am a fellow servant with you and with your brothers and sisters who hold to the testimony of Jesus. Worship God! For it is the Spirit of prophecy who bears testimony to Jesus.”

Angels, Saints and "ordinary" Christians: They all live in the union with Christ. We can ask any of them to pray for us, but the miracle would have to come from God.

_{¹ A metaphor comes to mind that may be inappropriate but is too good to let it pass: Angels and Saints are like lobbyists. We can ask them to intercede on our behalf, but they cannot perform official acts.}