Sorry but I'm really not convinced, though I am really enjoying this conversation so thank you for your reply.

Reading the article you shared, my impression is that if the medical clinic question is the inverted form of the previous sentence "sure, you do", then the inverted part is the "do" moving to the front of the question in "does your medical clinic?"

Responding to your examples, I feel the exact same way. They read completely unnaturally to me. Do you actually hear people speak like that? I don't think I ever have. It really sticks out to me because I would expect the context for 'do' to follow on, eg "but would your medical clinic do better?" I agree that a sentence like "I don't, but your medical clinic might do" is acceptable like in the original link you provided, but when posed as a question, I would expect to drop one of the words in "might do" ie "but might your medical clinic?" or "but does your medical clinic?"

Looking forward to hearing your thoughts.