In 有一只小鸟真可怜，它在树枝上冷得发抖 why is 有 at the start?
So here I assume 有 indicates the bird‘s existence

Correct. 有 is frequently used as the equivalent of &quot;there is&quot; in English. The typical construction is

place + 有 + noun
Example: 我家里 (place) 有 (existence) 很多椅子 (subject) -&gt; There's many chairs in my house

Then you can also omit the place to express just generic existence, as in your sentence:

有一只小鸟真可怜 (nowhere in particular) there's a little bird which is really pitiful

This construction can be productively used in story-telling, where the exact location of the subject is not very relevant. Same as in English.
If you add a time complement instead of a location, you get the classical story introduction:

很久以前有一只小鸡~ once upon a time (nowhere in particular) there was a small chicken...