Seems perfectly possible to me a bird could sense the difference between a vibrant, active person and a person close to death.
I once pitched a tent somewhere in the deep forests of Sweden, on a trip with my mother and my sister. I sat the tent up on some natural cushions of dry peat, so the ground was nice and soft to sleep on. As it became darker I suddenly heard a large CRACK about fifty metres away as a big twig was snapped in half. The sound was followed by the deep sniffing of air and loud stomping, so I popped my head out of the tent to take look.
Atop the closest hill was the King of the Forest (that would be a moose), looking like he was pondering some deep mystery while on his evening stroll, his mighty antlers brushing away big branches like they were toothpicks. He wasn't too worried about his surroundings until he noticed my scruffy old head. He stopped dead quiet and opened his eyes wide in surprise. Then he snorted heavily in disappointment, like he was muttering a curse, and then he shuffled his heavy body around and walked the other way.
I got the distinct sense that this proud being had a very sincere distaste for people...
Here is a far simpler explanation. Our brains are evolved to find patterns even when they aren't there. Look how weak the correlation is. His family simply discarded all the times the correlation failed to apply and remembered the one time the bird cried not on the day his grandfather died but merely that same year.
It’s far fetched maybe but dogs can smell Parkinson’s and cancer before the person develops medically detectable symptoms. So maybe not so far fetched.
Doesn’t have to be a pet. Doesn’t detect when the person will DIE of cancer, yet it shows up later.
Birds have been shown to return to the same locations if they fit their needs. Perhaps it was indicating an illness.
Far fetched but absence of evidence is not evidence of absence.
Statistics are not science. Science requires real world evidence and other animals offer at least some evidence they can detect fatal illness in advance.
Most birds and pheasants in particular have don't have as well developed a sense of smell as you do and aren't liable to be within 100 yards of people for long.
Even if they could smell impending disease why would they particularly seek out a high place to announce this and how on earth would you distinguish this from a call which announced it's desire to mate or warning about predators including yourself.
Birds make noise all the time. They can't smell you from far away ergo they can't smell impending death. It's like an ink blot test any meaning you are finding is smuggled in by you.
I’m not seeking meaning, I’m seeking falsifiable experiment to conclusively rule out the possibility.
No reason to assume similar can’t occur between species.
Perhaps the bird “read his mind” in such a way as to intuit something was breaking.
Perhaps applying simple cost/benefit analysis is too reductionist to reach a conclusion?
Which is my whole point.
You have fun falling into the abyss. I’m gonna go see what reality is made of.