Is there an incident in the Satcharita that shows Sai Baba's concern for someone who had stopped their family's traditional worship?

Yes, the story of Bhagwantrao Kshirsagar in Chapter 4 is a perfect example. Bhagwantrao's father was a devout worshipper of Vitthal, but after his father's death, Bhagwantrao neglected the family's traditions. The worship and food offerings to the household Vitthal idol ceased. When Bhagwantrao came to Shirdi, Sai Baba, who referred to his father as a friend, revealed His awareness of the situation. Baba said, "He never offers food (Naivedya); he keeps me hungry too. He keeps Vitthal hungry too, that’s why I brought him to Shirdi." Baba then stated his intention to remind Bhagwantrao and put him back on the path of worship, showing his deep personal interest in reviving a devotee's forgotten duties.