Does the Hadith "or a righteous child who supplicates for him" stipulate the death of a person for the righteousness of the child to be realized, and is what was mentioned in the publication considered a measure for identifying the righteous?

The Hadith (definition) does not mean that only a righteous child prays for their father; rather, it means that a righteous child is a deed for the father that does not cease with his death. The righteous child is singled out because he is likely to pray [for his parents], as he benefits himself and others. As for the wicked person, he may pray for his parent, but his prayer does not indicate his righteousness, because a servant's righteousness is determined by the predominance of good deeds in his life, and his prayer for his parents is an indication of his righteousness, but not a definitive proof of it.