Does the text from Chapter 36 confirm that Shirdi is Sai Baba's village?

No, the provided text from Chapter 36 does not mention Shirdi or any other specific geographical location as Sai Baba's village. On the contrary, it makes a more abstract, spiritual point about his omnipresent nature by describing him as one "who has no name or village, but possesses infinite glory." The passage focuses on his transcendental qualities, such as his ability to turn a beggar into a king and his detachment, rather than his physical ties to a particular place like Shirdi.