Someone who has a lot of empathy probably can't handle being a surgeon in the first place. It would be too stressful for them.

A terrible and disingenuous trope. Surgeons make numerous decisions during surgery that require extraordinary empathy to deliver the best outcome. And, unsurprisingly, surgery outcomes are strongly correlated with strong doctor-patient relationships.

Moreover, your use of the word "handle" is peculiar. Modern operating rooms are not horror movie sets.

(I apologize for my aggressiveness. This is a major pet peeve!)

You can make the correct decision for the patient by wishing for the best outcome for them. It does not mean you have to have empathy for them. Empathy is the ability to place yourself in another's position.

Let me give you another example. I'm an expert poker player. I understand why worse players play the way they do, but I can't empathize with it. I simply can't put myself in a position where I'd play so poorly, so I can only say I understand it from a "they're bad, so they make mistakes" point of view. But this does not preclude me from making thousands of dollars at the casino from said players. Someone who is able to better empathize with that person might also make more basic strategy mistakes than me and they won't profit as much.

Empathy does not guarantee better outcomes. A doctor could have a great relationship with a person and still not able to empathize fully with their issue.

or they have sufficient empathy to know that what they're doing might be painful but will ultimately save your life or increase your quality of life, and thus is absolutely worth doing.

That's not empathy, that's helping others. You can help a person without really feeling what it's like to be in their shoes.

And then your leg gets cut off at the wrong angle since his hands are not under his control due to emotional connection. or He doesn't cut it off and then your diseases or issue spreads and your whole body gets paralysed.

I doubt these are common failure modes.

Doesn't matter, this is plainly grasping at straws for the sake of argument.