I think that there was nothing explicitly preventing flying schools operating in the legislation. The issue was always the restriction of movement element within the law. Being outside the home to attend a flying lesson possibly could not be defined as a reasonable excuse. This section has now has been altered substantially, and the English law seems now only to restrict being away from home overnight (section 6).
Everyone seems to be waiting for guidance that is never forthcoming, as those who issue the guidance never want to commit and, I also believe, don't understand the law. That is why we see guidance saying that things are now permitted when under the law they had never been banned in the first place.
The only solution is to return to looking hard at the law.
Section 7 states that a reasonable excuse (for gathering) is if it,
"takes place at an educational facility and is reasonably necessary for the purposes of education."
It does not appear to define what an educational facility is, or what is reasonably necessary for the purpose of education. Note it says reasonably necessary for the purpose of education, not that that education must be reasonably necessary. This is a very important differentiation.
It is easy to see why CPL schools, are comfortable to restart, but by the same token there is nothing explicit saying that an organisation that is advertising itself as a flying school cannot educate (train) PPL students. Remember the law is written to prevent things as opposed to approve things; however it appears to be constructed here in a way that not only doesn't prevent flight training but approves of educational gatherings which flight training is not excluded from.