Does Christianity claims that congenital deaf-blindness could never be cured by humans?

I'm not going to focus on the what is/isn't a "son of God" debate. That would make a good second question. I'll simply leave you with a Biblical verse to consider: Acts 17:29, "Forasmuch then as we are the offspring of God, we ought not to think that the Godhead is like unto gold, or silver, or stone, graven by art and man’s device." In other words, we are the children of God, but that doesn't mean we should create God in our image. That's my two cents worth.

You asked the question,

So, preserving the un-ambiguity and distinction of Jesus, does Christianity claim that humans will never be able to cure congenital deafness/blindness?

As the link you provide indicates, there are many reasons why someone might be (for example) deaf from birth. I'd like to point out that your link is talking about a specific malady called "deafblindness," meaning that the baby is born both deaf and blind. However, I think we can consider that specific condition as one of the same class of conditions that reflect the miracles performed by Jesus.

A quick Internet search reveals that as of 2021 humanity still has no cure for congenital deafness (continuing to use it as our example), even though it is the most common birth defect in the U.S.

Congenital hearing loss is the most common birth defect, estimated to affect 2‐3 in every 1000 births. Currently there is no cure for hearing loss. Treatment options are limited to hearing aids for mild and moderate cases, and cochlear implants for severe and profound hearing loss. (Source)

The cited article discusses the known environmental and genetic causes of congenital deafness and points out the work toward finding a cure.

Over the last decade, next generation sequencing (NGS) has allowed the identification of multiple deafness related genes and pathways. Together with recently developed molecular biology tools, this has opened the field to the development of hearing loss therapies that include gene delivery and editing, stem cell transplantation, and targeted drug delivery systems. (Ibid.)

For the most part, these advancements will help humanity avoid congenital deafness. But that's not what Jesus did. His miracles healed the deafness in adults. He didn't avoid anything, He fixed it after the damage was done.

So, do we expect humanity to cure it in adults?

I suspect that you will find different responses from different Christian traditions. In the case of my tradition, the answer is, "we don't know if humanity will find a cure before the Second Coming of Jesus Christ, but it wouldn't surprise us if we did." We believe this due to several passages of scripture.

1 Cor. 12:8

For to one is given by the Spirit the word of wisdom; to another the word of knowledge by the same Spirit.

Knowledge is a gift of the Spirit. We believe the great strides in knowledge achieved by humanity since the resurrection of Jesus is due to this spiritual gift.

Matt. 24:24

For there shall arise false Christs, and false prophets, and shall shew great signs and wonders; insomuch that, if it were possible, they shall deceive the very elect.

and,

2 Thes. 2:9

Even him, whose coming is after the working of Satan with all power and signs and lying wonders.

And like any good gift from any loving parent, it can be used unwisely — even deceitfully. Therefore, we believe that as the day of the Second Coming of Jesus approaches, the many miracles He performed will be duplicated such that, "if it were possible, they shall deceive the very elect." We believe that the advancement of science will be one way these prophecies will be fulfilled.

2 Tim. 3:7

[The people of the Last Days will be] Ever learning, and never able to come to the knowledge of the truth.

Thus, we believe it is quite possible for humanity to discover a cure for congenital deafness, blindness, heart disease, etc. as a counterfeit miracle duplicating the miracles of Jesus and continuing to lead people away from the saving and exalting Truth that is in Christ Jesus.

We can all be children of God (see SE answer)*, Jesus is uniquely begotten of the Father.

Healing was/is not unique to Jesus. Jesus commanded his apostles to:

7 And proclaim as you go, saying, ‘The kingdom of heaven is at hand.’

8 Heal the sick, raise the dead, cleanse lepers, cast out demons. You received without paying; give without pay.¹

Which they did and we have accounts of throughout the New Testament (large number are recorded in Acts). The apostles were also considered ordinary men, see Acts 4:13

13 Now when they saw the boldness of Peter and John, and perceived that they were uneducated, common men, they were astonished. And they recognized that they had been with Jesus.

There is also nothing in the bible stating that humans can't cure something through medical means alone.

_{* some denominations have different beliefs on who is considered children of God, but I believe most could agree that Christians would fall under this category}

_{1 Matt 10:7-8}

If I understand the question correctly the answer is "no." Other people than Jesus (as the Son of God) can cure congenital blindness or deafness. Also medical science will probably be able to cure it son, perhaps has already done so in some cases.

The Old Testament provides many examples in which people were cured of major diseases by people other than Jesus. To name just a few:

• 1. Numbers 16:46-50 - Aaron stopped the plague which had killed 14, 700 people by offering incense and making atonement for the people.
• 1 Kings 17:17-24 - Elijah raised a widow’s son from death.
• 2 Kings 4:8-17 - Elisha granted a child to the formerly barren Shunnamite woman.
• 2 Kings 13:21 - A dead man was thrown into Elisha’s tomb and contact with Elisha’s bones raised the man to life.
• 2 Kings 20:1-7 - Hezekiah contracted a terminal illness and prayed for healing. Isaiah received a word from God that he would live for a further 15 years

If people other than Jesus could cure the sick and raise the need, there is no reason to presume they could not also cure the blind and the deaf. Of course, some Christians may attribute these miracles to Jesus as God's eternally pre-existent son, but the OT does not say so.

In addition recent scientific advances provide evidence that congenital blindness may be cured by medical science such as gene therapy. As far as I know there is nothing in the Bible that would make it necessary for Christians to deny such cures. So no, Christianity does not claim that those who are blind or deaf from birth cannot be cured by humans.