There's a BIG difference between emergency triage and the hospital ward.
You are concerned with one character who is in the process of dying. Injury/Illness is compromising their ability to live. With this level of care, you need to be 100% alert and know as much about the patient at all times. This is different from getting cut and needing a few stitches. Treat Deadly Wounds is a character's Aorta is nicked/cut. They are bleeding out. You need both hands on the wounded character, your attention to keep them awake/alert (so they don't pass out/go into shock) and frequently you need someone else to pass you stuff as necessary.
This is a less stressful task. If you have ever seen a doctor make rounds in a hospital/hospital show on TV, this is long-term care. The doctor has a chart to consult, takes a few minutes with each patient, then moves on. In a Pathfinder setting, you are likely to take care of a ward of patients, check/change bandages, apply poultices, etc. You don't need to hurry generally, and when you do, the other patients can wait.
With that in mind, you can either perform Treat Deadly Wounds on one person, or Long Term Care a recovery ward. The only exception would be if you are doing Long Term Care on a ward of stable patients and a new one comes in needing Treat Deadly Wounds. Then you administer emergency care to the one patient and then once stabilized count that person as part of the ward when you go back on rounds.