The idea of banning private prosecutions is without doubt or exception the most stupid suggestion that I have ever read.
A "private prosecution" is literally any prosecution launched by anybody other than the state. Let's say I suggest that you are a child molester and get a national newspaper to print these deliberately false accusations which causes you to lose your job.
You would no longer have any ability to use the criminal justice system to gain recourse by suing me because that would be a "private prosecution" which you'd banned.
You would have effectively have near totally abolished the rule of law. Contracts would no longer be enforceable. If your employer decides he's not going to pay you anymore? You can't take him to court and demand payment. You supply goods and somebody refuses to pay? No recourse.
The issue with these prosecutions is not that the Post Office were allowed to bring these prosecutions, it is that the people responsible for the prosecutions have behaved illegally, and the solution is to apply the law.
Criminal Procedure and Investigations Act 1996
(1)The prosecutor must—
(a)disclose to the accused any prosecution material which has not previously been disclosed to the accused and which might reasonably be considered capable of undermining the case for the prosecution against the accused of assisting the case for the accused , or
(b)give to the accused a written statement that there is no material of a description mentioned in paragraph (a).
So here is a simple case; the prosecution did not comply with A, and they committed perjury in providing the statement at B. They have broken literally dozens of laws and gratuitously abused every underlying principle of how the Criminal Justice System works by their misconduct.
The solution to this is not abolishing the law, it is to prosecute every person who was involved to the maximum extent possible of the law, and in every case to hit them with the maximum possible penalties.