Why can a bird be pulled but never caught?

I think it's probably just because in British slang pull has that meaning independently of bird. From OED...

pull: trans. 12a: Brit. slang. To pick up (a partner), esp. for sexual intercourse; to seduce. Also intr.

It also occurs as a noun in the expression on the pull, and there's no reason why a couple of young British men shouldn't hope to pull some girls on a night out. Come to that, the girls they end up with may have gone out hoping to pull some blokes.

As to why young British women are called birds, OED says it derives in part from a now-obsolete

burd: a poetic word for ‘woman, lady’; the female counterpart of berne n.;
in later use chiefly = ‘young lady, maiden’.
berne: a warrior, a hero, a man of valour;
in later use, simply one of the many poetic words for ‘man’.

Of chick, OED says applied to human offspring; = chicken n.; esp. in alliteration with child. Sometimes as a term of endearment, with citations starting from 1320. But their earliest citation for the current (well, hopelessly "dated", imho) sense girl; young woman. slang (orig. US) is 1927.