There would have to be different "levels" of qualification, as there is in, say, the medical profession, so that the actual qualification anyone claims is restricted to some degree that is understandable by the general public.
My GP is a qualified doctor, but he's not "qualified" to do surgery as that requires further qualifications. A surgeon with all the additional required qualifications might be ok to stand -in for a GP but not vice-versa. However, many specialist surgeons concentrate on particular areas of expertise, so, for example, a surgeon who concentrates exclusively on hip replacements isn't going to do cataract operations and so on. That's why the medical profession has so many levels, so if a doctor has passed extra exams and done all the studying and clinical practice he can then do the surgery for which he is "qualified". He would probably be useless as a GP though! However, he is "qualified" to do so, as he is over-qualified, but he probably wouldn't have any interest in that post, and should he retire from specialst surgery and seek a post as a GP in the years immediately prior to retiring, he will need additional specialist training just to cope with the essentials of dealing with every patient who crosses his door and also training to deal with the prescribing and paperwork etc. This does not mean he will have the "bedside manner" a good GP needs, but he will have a good knowledge of general medicine, plus the NHS backup for assistance when required.
The medics have clearly defined levels, and they are tested at each level. Electricians do not have these prescribed levels, only a few restrictions designed to ensure some level of competence.
If I pass an exam that shows I am competent to install, inspect, test and certify in domestic installations, I am "qualified"...but unlikely to have a clue about installing more complex features which truly require more teaching and hands-on experience.
All well and good...but who is going to set out the rules for these levels, who is going to teach these courses, where and at what cost?
No new law that bans unqualified people from doing electrical work is going to work unless the training and testing infrastructure is in place, because without those elements there is no way someone can become "qualified".
Even if colleges had such courses in place, who is to determine what level of "qualification" is required for the teachers of these courses?
Yes, I know there are some "levels" in place but they are poorly defined, and it's clear that those seeking to become proper electricians have difficulty in getting training for hands-on experience etc.
Even if there were proper courses, like some of the colleges do provide at present, places are limited and it takes a long time to cover all that is required in theory and in practice, and many will be unable to afford the cost.
Assuming there were to be a "basic" course so that one could become a proper, qualified, domestic installer, that person would have to complete another year or two at college and in the field before moving up a level, all of which is extremely difficult for a working electrician trying to make a living and support a family. I could go on...