It’s definitely the same thing. We can test this using other modals and auxiliaries in equivalent question constructions to show that we’re dealing with analogous structures:
If making a question with “might”, for example (with the pro-predicate base sentence “But your medical clinic might do”), we get “But might your medical clinic do?”
With “would”, “But would your medical clinic do?”
So, with “dummy do”/do-support leading to the insertion of “do” for inversion purposes, along with the separate pro-predicate “do” lower in the clause, “Your medical clinic does” (or possibly “Your medical clinic does do”) becomes in the same way “Does your medical clinic do?”
It’s definitely the same thing. We can test this using other modals and auxiliaries in equivalent question constructions to show that we’re dealing with analogous structures:
If making a question with “might”, for example (with the pro-predicate base sentence “But your medical clinic might do”), we get “But might your medical clinic do?”
With “would”, “But would your medical clinic do?”
So, with “dummy do”/do-support leading to the insertion of “do” for inversion purposes, along with the separate pro-predicate “do” lower in the clause, “Your medical clinic does” (or possibly “Your medical clinic does do”) becomes in the same way “Does your medical clinic do?”