> I do know all that, but it still doesn’t really seem enough to qualify them as the most important person.

A surgeon can perform a procedure on a living patient or a cadaver and whichever is the case, it does not affect their work. An anaesthesiologist is responsible for ensuring the former does not become the latter, excluding catastrophic surgical procedure failure.

> Remove the anæsthetist, and procedure forbids you from continuing. Remove the surgeon, and it’s impossible to continue.

The opposite is the case (see above).