You guys are assuming that every place adheres to a "one service, one attending" model. So idealistic.
To be devil's advocate:
My residency program (surgery) did not work like this. There were multiple surgery services. On any given service, there were multiple attendings with patients at all times. If service A had 24 patients, 12 might be from attending S, 6 from attending T, 4 and 2 each from attendings U and V. Each attending managed only their own patients and each staffed patients with the residents during the day. Residents on Service A may cross cover Service B at night, which might be 8 attendings total (or more). But the attending on call may be a Service C attending. Keep in mind that if an emergency came up with a given patient, their own attending would be contacted, NOT the attending on-call, unless the patient was in the ER and unassigned. If a resident from Service A is sick, what attending covers? Do all 8 attendings on Services A and B come in and cover their own patients? Does the on-call attending from Service C take in-house call covering Service A and B and manage other attending's patients, but not cover his own service as "first call"? Keep in mind the resident is covering multiple services and multiple attendings. Now throw in politics typical in academics where certain attendings may not like each other OR simply have vastly different approaches/subspecialties, etc. etc., and now you have a real mess as to how this could work. Add in the fact that the on-call attending does not get paid (and cannot charge for) to manage another surgeon's post op patients who are in a global surgery period if they are part of the same group. So does this attending then need to get paid by the hospital for "call" (more expensive than a resident) or are they working for free (if so, why should or would an attending agree to this)? It becomes far more complicated than switching the residents around or making residents cover even more patients each overnight to compensate for a missing resident.
For community hospital rotations, some guys in private practice (especially surgical specialties) cover multiple hospitals and may be unable to stay in house at one place all night to cover a sick resident if something comes in at one of the other hospitals they are covering, especially if the other hospital (which likely doesn't have residents) pays them to take call AND they can bill for the encounter.
I'm not in academics, btw. Just trying to point out how difficult it is to cover a sick resident. No matter what, somebody still gets screwed over...