I’m a strong proponent for listening to your gut, but that’s very different from a bird landing on your home predicting death, especially when the correlation is weak (sometime later that year). It’s easy to attribute something to something else if there’s an indeterminate timeframe in between.

Remember that people see what they want to see in noise too. Clouds often look like something tangible, that doesn’t mean it’s a sign. If I squint, I can see shapes in pretty much any random pattern. And people tend to remember events that they attribute meaning too (in this case the old wives tale about the birds and then the family member dying) and forget the events that have no meaning to them (how many times did the person see said birds without anything happening and just forgetting about it?)

Nobody has yet managed to prove the existence of the supernatural, but many people have tried. That doesn't mean it doesn't exist, but it seems pretty unlikely, especially when other better-understood causes exist.