mini-moan: past tense possessive

This is something that has bugged me for years. Nowadays it is no longer correct to say or write 'he' when talking about an individual of unknown gender. Previously 'he' would often be used as a general catch all just as 'man ' is used to refer generically to humans.

Due to a change in gender roles and a move to equality 'he' has gone and fair enough, but it has been replaced with the monstrous 'he/she', 'he or she', 'his/her' .etc. And this is what I cannot understand, instead of saying he/she which is clumsy and painful just say 'they', 'their', 'them'.
Our language has built into it a gender neutral way referring to individuals or groups of unknown or indeterminate gender, so why do people insist on 'he/she'.

I can only assume that people are naturally writing or saying 'he' and then quickly correct themselves to say 'he/she'.

And why the feck is a cat always 'she' and a dog always 'he'. If another person asks 'What's his name' when referring to my female dog I'll deck them.