Can you provide an example from the Shri Sai Satcharitra where Baba verbally repeated a devotee's private thoughts?

Certainly. Chapter 28 of the Shri Sai Satcharitra recounts a striking incident involving Laxmichand. During the Chavadi procession one night, Baba developed a cough. Laxmichand, upon seeing this, thought to himself that the cough might be the result of someone's "evil eye." This was purely a mental thought. The very next morning, when Laxmichand was in the masjid, Baba spoke to Madhavrao, saying, "Yesterday I had much coughing; is this the effect of an evil eye?" This event stunned Laxmichand, who realized Baba had perfectly echoed his private thought, leading him to conclude that Baba truly dwells in everyone's heart and possesses profound inner knowledge.