The verse people use, and what it says
Protect yourselves and your families from a Fire. That verse gets produced to justify severity with children, so it is worth reading what it actually instructs.
The verb is protect. Not punish, not correct, not discipline. And look at the order in the verse. Yourselves first, and then your families. The parent is the first object of the instruction, not the enforcer of it.
Qur’an and Sunnah
يَٰٓأَيُّهَا ٱلَّذِينَ ءَامَنُواْ قُوٓاْ أَنفُسَكُمۡ وَأَهۡلِيكُمۡ نَارࣰ ا وَقُودُهَا ٱلنَّاسُ وَٱلۡحِجَارَةُ عَلَيۡهَا مَلَٰٓئِكَةٌ غِلَاظࣱ شِدَادࣱ لَّا يَعۡصُونَ ٱللَّهَ مَآ أَمَرَهُمۡ وَيَفۡعَلُونَ مَا يُؤۡمَرُونَ
Believers, guard yourselves and your families against a Fire fuelled by people and stones, over which stand angels, stern and strong; angels who never disobey God’s commands to them, but do as they are ordered
Qur’an 66:6
Translation: M.A.S. Abdel Haleem, The Qur'an, Oxford University Press, 2004.