It sounds brutal to me too, and I personally am against it. Stoning, in general, is extremely rare in official practice:
Out of the world’s forty-nine Muslim-majority states, six retain the punishment in deference to Islamic legal tradition, despite the fact that the Koran, unlike the Bible (Deuteronomy 22:24), does not mention it. Of these countries only Iran, which officially placed a moratorium on stoning in 2002 but still gives leeway to individual judges, has actually carried it out. -- How She Wants to Modify Muslims, Max Rodenbeck
Outside of terrorist organizations, it's hard to find support for stoning.
Stoning is not mentioned in the Qur'an; not even the Islam-critical website WikiIslam contradicts that.
The section on "Sodomy" in Major Sins by Al-Dhahabi, which is generally a comprehensive source, does not mention stoning (even in the subsection "The Punishment of Sodomites").
It emphasises that sodomy was the sin, not homosexuality per se:
Sodomy is also approaching a woman from the anus.
and quotes the hadith:
Allah does not look to a man who commits sodomy with a man or a woman. (Reported by At-Tirmidhi. It is good but Gharib hadith)
A large portion of this section is focused on preventing harm to minors:
AI-Hasan Ibn Zakwan said, "Do not be accustomed to sit with the wealthy men's kids since they are good looking like the virgin ladies. They might be more fascinating than women.”
The closest mention is this:
It was narrated from Ibn`Abbas that the Messenger of Allah (ﷺ) said:
Whoever you find doing the action of the people of Lut, kill the one who does it, and the one to whom it is done. [Grade: Hasan (Darussalam)] (sunnah.com; and another version)
This could be interpreted as implying the hudd punishment for sodomy, which one might argue implies stoning.
There is a weak hadith not mentioned in Major Sins:
Narrated Abdullah ibn Abbas:
If a man who is not married is seized committing sodomy, he will be stoned to death.
Abu Dawud said: The tradition of 'Asim proved the tradition of 'Amir b. Abi 'Amr as weak. -- Sunan Abi Dawud 4463 (sunnah.com)
Wikipedia summarizes the punishment for sodomy in an LGBT context:
Outside of the Quran, there were varying opinions on how the death penalty was to be carried out for such sexual transgressions. Abu Bakr apparently recommended toppling a wall on the evil-doer, or else burning alive, while Ali bin Abi Talib ordered death by stoning for one "luti" and had another thrown head-first from the top of a minaret—according to Ibn Abbas, this last punishment must be followed by stoning. However, according to Abu Hanifa, homosexual acts are not hadd, but ta'zeer (maximum 39 lashes).
Even proponents of punishment for sodomy disagree on the type of punishment.