It's not just flight training that needs a PT CofA (or, when CofA are no longer issued, maintenance to the PT CofA regime).
It's any kind of renting, including the case of a plane owned by a limited company of which you are a 100% owner and the plane being rented back to you by the company. Obviously the actual operation is identical to you owning the plane directly. The technical rental is done to avoid getting hit for the Inland Revenue Benefit in Kind, if any portion of the flying is for private purposes.
One solution to the above situation is to go N-reg, where this isn't an issue.
IMHO the CAA have gone way over the top in trying to screw down the options for freelance cowboys doing flight training out of the back of a Ford Escort. Years ago, a CFI explained to me once how he proposed to run a school using Private CofA planes, by making each student a Director (or something like that). Later, he vanished after embezzling a load of money from the school. One could of course ask why on earth does GA end up with so many crooks and cowboys. It's at least as bad as double glazing / central heating / loft insulation / cavity wall insulation has been, in their respective darkest days. That's why we have the stupid incomprehensible ANO wording, the stupid rules about not advertising for passengers (PPL cost sharing) except on a noticeboard in a club, etc.