It seems like a lot of coincidences led to Madhavrao showing up just when Dixit needed him. Was this just luck, or is there a deeper explanation in the text?

The text strongly suggests that the events were not mere coincidence but a compassionate response to a devotee's strong desire. Chapter 51 explicitly contrasts human plans with a divine one, stating, "While humans make their plans, God has another plan." It notes that as Dixit sat worried, the "Almighty felt compassion" seeing his devotee's strong desire. Madhavrao's sudden trip to Nagar was prompted by a telegram about his mother-in-law, who, upon his arrival, was found to be in fine health. This detail reinforces the narrative that his arrival was a pre-ordained miracle arranged to fulfill Dixit's heartfelt wish to be taken to Sai Baba.