Interesting video but I think the person being questioned could have stood up for Part P a little more confidently.
It's true, the work can be carried out by anyone but it still has to be signed off by someone who is registered. To be registered you have to be assessed as competent by a scheme. My scheme (Stroma) are quite strict and would not pass me unless I met all their criteria, answered all there questions and carried out my assessed job safely.
The registration scheme is far from perfect but I do think we just enjoy saying how bad they are as we feel bitter at having to fork out £300-£500.
I think one possible answer would be to police it more stringently, but then we'd have to pay even more!
Also, I think shoddy work is more often carried out by those who just don't care about their standard of work rather than if they have qualifications or not.