I work in the music industry, so without getting caught up, as i can appreciate both sides of the coin, Ill answer a few quiries!
How do they know what music is played & pay appropriate people, Simple, Neilsen tracking, A computer that listens to all Radio stations & can recognise 99.99% of tracks.
PRS is for the UK, Almost all Western Countries have similar organisations, were similar rules apply, its not just the UK
The money has to go to lots of people, Writers, Producers, Performers, Record Labels etc