Without wishing to get too bogged down, the link you've provided shows just what a mess the law around private prosecutions is.
As you say, private prosecutors are not bound by the Code for Crown Prosecutors, and it is this Code which sets out the formal evidential and public interest tests.
However private prosecutors
are still bound by the Criminal Procedure Rules 2020 which includes some public interest and evidential considerations.
So we're both sort of right here. I'm right in that they should. You're right in that they often don't and, as they're not subject to the formal Code, there's no comeback if they don't.
The current situation is that they're supposed to consider the public interest in bringing a prosecution but there is no meaningful sanction if they don't. So the TOCs don't. As the government notes,
"where this separation [between prosecutor and victim] is not observed, there are risks that the prosecutor brings a case without meeting the evidential and/or public interest test, and instead because it is in the organisation’s interests to do so."
As always, why do something that will take time and cost money if there's no sanction for not doing it?
I'm not a solicitor, I'm not an expert, and I defer to the advice of people who are solicitors or barristers. I do, however, hold the same professional qualification that the TOC prosecution staff also seemingly hold and use. And my training for that qualification was that I should follow the Code even though, technically, there are no consequences if I don't.
Personally, I'd make it so that all private prosecutions have to be signed off by a senior Crown Prosecutor, and that by itself would put a stop to all this nonsense in a flash.