> From what I heard, this provision for public prosecution dates from the olden days when the post office was carrying important and sensitive messages, like that from the royals, and when they had to fight highway robbers and such.

You were misinformed.

> As for the argument, "nothing else done otherwise", we don't see similar prosecution privileges for other essential services in life (health service, public transport etc.).

Yes we do; as the article says, in the UK any private organisation or individual has the right to bring a prosecution.