As someone pointed out above, false accounting and fraud charges have nothing to do with SJP.
Private prosecutions have long been a feature of English law. The Prosecution of Offences Act 1985 specifically reserves that possibility (although the CPS can take over prosecutions). Private prosecutions can also be used to good effect, such as by the RSPCA in animal welfare cases, although the competing benefits/disadvantages of private prosecutions are a valid legal question see this
video for instance. Whatever is to be made of it, it should be noted that private prosecutions have a wide range of uses (often in niche, specialised fields) which should each be considered before abolishing them.