To put it simply, Xu Lei got pregnant at a dangerous age. Her children were weak, and she gave birth to twins. Xu Lei couldn't take care of three children at once.
They wanted to announce that their eldest son was adopted once he was born. But now that she had two infants, adopting one in such a hurry would arouse suspicion.
If she didn't have her children, it would be fine if the eldest was seen as an adopted son. But when she had children of her own, saying that the eldest son was adopted would hinder his growth.
Since they were family, they wouldn't treat the eldest son differently. But some trolls just loved to fabricate stories. Heavens knew how much that would affect their family.