While you are technically correct it's not actually the law in the UK . The PRS are the people who will come round and check (not the BPI) and they're not so much a licensing authority as a club that most of the artists, bands, labels etc are members of. It's basically a protection racket dressed up in shiney clothes. If you limited your musical output to non-PRS-registered artists then TECHNICALLY you wouldn't need a PRS license. Technically is in capital letters, because you are little people and they are loaded.
One fun fact is that if you are a PRS artist and you perform your own work live in a PRS registered venue then you get paid from the PRS's coffers because on that date $venue broadcast your music. Even if the venue ALSO pay you in person for your performance.
So if you want to get rich quick, make some music, register it with the PRS then sit in a PRS registered pub all day playing your own music to the lonely bloke and dog.