To some extent that question is already answered in practice on the ground because psychiatrists are not involved in all competency assessments, and that includes ones that have life-and-death consequences.
For example, in oncology, if a patient wishes to decline chemotherapy or their treatments, it does not require a psychiatrist to always be involved. If the oncologist feels that it goes beyond their ability—and when I say this, it is not in a pejorative way—or if the oncologist is uncertain whether they're getting the whole picture and whether, perhaps, there are other influences or mental illnesses present, then they can pull in the psychiatrist. It's not something that is routinely done by default.