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.