Yes, God does NOT make mistakes.

It's a true saying, and It's Biblical. It does not appear anywhere in Scripture in any English translation of the Bible, but there are verses like Ecclesiastes 3:11, Psalms 18:30, 2nd Samuel 22:31, and Deuteronomy 32:4 that echo this sentiment. And there are verses like 2nd Corinthians 12:9 which encourage us to continue despite suffering and 1st Peter 5:10 where He promises us an end to suffering.

But I see this saying used a lot to discredit or de-legitimize trans Christians' choices to seek medical or surgical treatment for their gender dysphoria, and it's been said to me too.

I can agree with the original sentiment just as a statement of fact because it's Biblically grounded. God does not (and cannot) err. However, in this world we do have tribulation. (John 16:33) Diseases exist because God allows them to exist. Cancer exists because God allows it. Mental illness exists. And so does gender dysphoria.

God also enabled men and women to discover cures for diseases, to invent drugs and therapies to heal mental illness, and to devise treatments that lead to cancer regression. Whether they be congenital, completely accidental, caused by other people's malice, recklessness, or neglect, or even self-inflicted, God does not fault human beings for trying to heal the sick and reverse debilitating health conditions and injuries, even the ones that come from poor decisions such as type 2 diabetes, heart disease, high blood pressure, and obesity. Rather, I think that God sees the practice of medicine and the mental health professions as human beings pursuing the second great commandment "You shall love your neighbor as yourself."

God may expect us to endure suffering when it would be spiritually fruitful to do so (such as the kind of suffering that arises out of persecution). But I do not think He expects us to endure suffering that has no meaningful purpose. He also condemns maltreatment and abuse of other people: do not mistreat or oppress the widow, orphan, foreigner, or the poor (Zechariah 7:10); do not make fun of deaf or blind people (Leviticus 19:14).

Yes, it is true that God does not make mistakes. But I also think every trans Christian's situation is different, and each of us needs to pray for God to show us what we should do or not do about our struggle with gender dysphoria as we have come to understand it, whereby our suffering has come as a result of the curse of sin. If He has not shown it to you that your gender dysphoria is something that you need to endure through the receiving of His grace and the pursuit of spiritual discipline, then, IMO, the minimalist extent of treatment that would bring you relief would be in keeping with the practice of Christian liberty according to one's clear conscience.

Bearing in mind also what Paul says in Galatians 5:16-26 (LSV):

And I say, walk in the Spirit, and you may not fulfill the desire of the flesh; 17 for the flesh desires contrary to the Spirit, and the Spirit contrary to the flesh, and these are opposed to one another, that the things that you may will—these you may not do; 18 and if you are led by the Spirit, you are not under law. 19 And the works of the flesh are also evident, which are: adultery, whoredom, uncleanness, licentiousness, 20 idolatry, witchcraft, enmities, strife, jealousy, wraths, rivalries, dissensions, sects, 21 envyings, murders, drunkenness, reveling, and such like, of which I tell you before, as I also said before, that those doing such things will not inherit the Kingdom of God. 22 And the fruit of the Spirit is: love, joy, peace, long-suffering, kindness, goodness, faith, 23 meekness, [and] self-control. Against such there is no law; 24 and those who are of Christ Jesus have crucified the flesh with the affections and the desires; 25 if we may live in the Spirit, we may also walk in the Spirit; 26 let us not become vainglorious—provoking one another, envying one another!