This is probably a very stupid question

No she isn't. We saw other Doctors on the ship - for instance in "The Schizoid Man" we see a Vulcan Doctor, Sela - played by Suzie Plakson, who would later play Worf's half-Klingon girlfriend, K'Ehleyr, as well as a couple of other roles in other Trek shows. In "Violations" we met Doctor Martin. In "Genesis" a Doctor Hacopian is referenced. In "Remember Me", Doctor Hill.

Those are the medical doctors we see/hear of, there are probably a good number that we don't. Given the ship's role in disaster relief and such, I'd guess there were at least a dozen doctors on board.