How can one be convinced that the children of non-Muslims deserve the fate of the Fire, considering that their parents' upbringing determined their religion and behaviors, and they had no say in that?

If the children of disbelievers die after reaching the age of accountability and remain in their disbelief, then they deserve the Fire for their actions. As for those who die before reaching puberty, scholars have differed concerning their fate. It is obligatory to believe that Allah's disposal of His dominion is just, and He is not questioned about what He does, for He is the Most Merciful of the merciful. This is what the Muslim's creed necessitates.