I don’t see a parallel between Hatzalah and doctors, at all.
Hatzalah is specifically designed for pekuach nefesh situations. Especially so on Shabbos, when Yidden are even more reluctant to call Hatzalah if it isn’t a possible real emergency (even slightly). Additionally, Hatzalah is designed for Yidden. It is true that some non-emergency use of Hatzalah is made, incorrectly. But Hatzalah isn’t advertising itself for non-emergencies or for non-yehudim. Those outside-its-scope uses take their use of Hatzalah on their own head/achrayis. Obviously Hatzalah can’t make a determination everytime a call comes in whether it is an appropriate use of Hatzalah, since they risk making a wrong call if they deny service and it turns out it really was a true emergency.
On the other hand, doctors are specifically there to provide medical assistance even when it is clearly not an emergency or life threatening situation. I don’t think an intern or resident expects to not provide medical assistance on Shabbos for people who come to his medical facility on Saturday even though they could wait until Sunday or Monday, without any real problem, to get their issue looked at. Additionally, most interns, residents and doctors are mostly dealing with non-yehudim. And, furthermore, unlike Hatzalah members who only respond on Shabbos after an emergency call comes in, the doctors are going (driving) to their place of work on Shabbos before any emergency occurs. Many Saturdays they might not even have any pikuach nefesh situation that whole day at their facility.